Twelve Tales in Mathematical Physics: An Expanded Heinemann Prize Lecture
TTWELVE TALES IN MATHEMATICAL PHYSICS: ANEXPANDED HEINEMANN PRIZE LECTURE
BARRY SIMON , Abstract.
This is an extended version of my 2018 Heinemannprize lecture describing the work for which I got the prize. Thecitation is very broad so this describes virtually all my work priorto 1995 and some afterwards. It discusses work in non-relativisticquantum mechanics, constructive quantum field theory and statis-tical mechanics.
Contents
0. Introduction 21. Summability of Divergent Eigenvalue Perturbation Series 22. Complex Scaling Theory of Resonances 113. Statistical Mechanical Methods in EQFT 184. Thomas–Fermi Theory 245. Infrared Bounds and Continuous Symmetry Breaking 316. N –Body quantum mechanics 447. Magnetic Fields in NRQM 788. Quasi-classical and Non–quasi-classical limits 859. Almost Periodic and Ergodic Schr¨odinger Operators 10110. Topological Methods in Condensed Matter Physics 12011. Anderson Localization: The Simon-Wolff Criterion 13112. Generic Singular Continuous Spectrum 14213. Further Remarks 150References 153 Date : November 26, 2020.2010
Mathematics Subject Classification.
Primary: 81Q10, 81T08, 82B24; Sec-ondary: 47A55, 81Q15, 81Q20, 81Q70, 81T25, 81U24.
Key words and phrases.
Simon, Schr¨odinger operators, quantum mechanics,quantum field theory, statistical mechanics. Departments of Mathematics and Physics, Mathematics 253-37, CaliforniaInstitute of Technology, Pasadena, CA 91125. E-mail: [email protected]. Research supported in part by NSF grants DMS-1265592 and DMS-1665526and in part by Israeli BSF Grant No. 2014337. a r X i v : . [ m a t h - ph ] N ov B. SIMON Introduction ? (cid:104) s0 (cid:105) ? The citation for my 2018 Dannie Heinemann for MathematicalPhysics reads: for his fundamental contributions to the mathemati-cal physics of quantum mechanics, quantum field theory, and statisticalmechanics, including spectral theory, phase transitions, and geometricphases, and his many books and monographs that have deeply influencedgenerations of researchers . This is very broad so I decided to respondto the invitation to speak at the March 2018 APS which says the talkshould be preferably on the work for which the Prize is being awarded ,by discussing the areas of my most important contributions to mathe-matical physics. I couldn’t say much in the 30 minutes allotted to thetalk so it seemed to make sense to prepare this expanded Prize Lecture.I will discuss 12 areas in theoretical and mathematical physics. Thefirst seven involve areas where my work was largely done during myPrinceton years, 1969–1980 (a kind of golden era in mathematicalphysics [675]) and the last four my Caltech years, 1980–1995 (I’ve re-mained at Caltech since 1995 but my interests shifted towards spectraltheory of very long range potentials and of orthogonal polynomialswhose connection to physics is more remote). The eighth area is onewhere I had work both before and after I moved to Caltech. It’s a plea-sure to thank Leonard Gross, George Hagedorn, Svetlana Jitmirskaya,Israel Sigal TKfor feedback on drafts of this article.x-ref? Many of the topics I’ll discuss have spawned industries (as shown bymy current Google scholar h-index of 114); any attempt to quote all therelated literature would stretch the number of references far beyond thecurrently 744 so I’ll settle for quoting relevant review articles or bookswhere they exist or perhaps focus on a very small number of laterpapers that I find particularly relevant.1.
Summability of Divergent Eigenvalue PerturbationSeries (cid:104) s1 (cid:105) Eigenvalue perturbation theory depends on formal perturbation se-ries (aka RSPT for Rayleigh-Schr¨odinger Perturbation Theory) intro-duced by Rayleigh [521] and Schr¨odinger [554]. The core of the rigoroustheory about 1970 when I began my research in this area were resultsof Rellich [531], extended by Nagy [479] and Kato [350] (see my re-view [682] of Kato’s work written on the centenary of his birth) andsummarized in Kato’s magnificent 1966 book [363].The Kato–Rellich theory in its simplest form considers operator fam-ilies A ( β ) = A + βB (1.1) ? ? WELVE TALES 3 where A and B are typically unbounded self–adjoint operators ( B need only be symmetric; see [680, Chap. 7] for a presentation of thelanguage of unbounded self–adjoint operators) on a Hilbert space, H .One demands that there are a and b so that D ( B ) ⊃ D ( A ); ∀ ϕ ∈ D ( A ) : (cid:107) Bϕ (cid:107) ≤ a (cid:107) A ϕ (cid:107) + b (cid:107) ϕ (cid:107) (1.2) In Kato’s language, one says that A ( β ) is type A perturbation of A .The big result of their theory is (cid:104) T1.1 (cid:105)
Theorem 1.1. If A ( β ) is a family of type A and E is an isolatedeigenvalue of A of finite multiplicity, (cid:96) , then there exist (cid:96) analyticfunctions, { E j ( β ) } (cid:96)j =1 , near β = 0 which are all the eigenvalues of A ( β ) near E when β is small. Moreover, there exist an analytic choice { ϕ j ( β ) } (cid:96)j =1 of eigenvectors, orthonormal when β is real and small. TheTaylor coefficients of E j and ϕ j are given by the Rayleigh–Schr¨odingerperturbation theory. Remarks.
1. For textbook presentations of this theorem, see Kato[363], Reed–Simon [528] or Simon [680, Sections 1.4 and 2.3].2. The
Kato—Rellich theorem assets that for β ∈ ( − a − , a − ), onehas that A ( β ) is self–adjoint on D ( A ).3. The theory is more general than the self–adjoint case. It sufficesthat A is closed, that B obey (1.2) and E be a point of the discretespectrum (isolated point of finite algebraic multiplicity). In that case,one has analyticity in the non-degenerate case ( (cid:96) = 1) but, in general, E j may be one or more convergent Pusieux series (fractional powers in β )).4. It is also not necessary that the β dependence be linear; a suitablekind of analyticity suffices,While this is elegant mathematics, the striking thing is that it doesn’tcover many cases of interest to physics. Perhaps, the simplest exampleis A ( β ) = − d dx + x + βx (1.3) the quantum anharmonic oscillator. This is the usual textbook modelof RSPT because the sum over intermediate states is finite and one cancompute the first few terms in the RSPT by hand.Moreover, it can be regarded as a toy model for a ϕ –field theory.Indeed, if one specializes a ( ϕ ) d +1 QFT in d space dimensions to d = 0,one gets a path integral for the A ( β ) of (1.3) and the RSPT terms canalso be written in terms of Feynman diagrams, at least for the groundstate (see, for example, Simon [595, 661]). B. SIMON
In this regard a celebrated argument of Dyson [163] is relevant. Henoted that quantum electrodynamics (QED) isn’t stable if e < e , heargued the Feynman perturbation series must diverge. Similarly, (1.3)for β < ±∞ - see, for ex-ample, [680, Theorem 7.4.21]). In fact, various estimates [326, 60, 581]show that the perturbation coefficients for the Rayleigh–Schr¨odingerseries of (1.3) grow like n !. (see the further discussion below).Two other standard models to which RSPT is applied are the Starkeffect in Hydrogen( indeed the title of Schr¨odinger’s paper [554] wherehe introduced his version of RSPT is “Quantization as an EigenvalueProblem, IV. Perturbation Theory with Application to the Stark Effectof Balmer Lines”) A ( β ) = − ∆ − r + βz (1.4) and the Zeeman effect in Hydrogen A ( β ) = −
12 ∆ − r + β x + y ) + βL z (1.5) The z in (1.4) and x + y terms in (1.5) are clearly not bounded atinfinity by A (0) (i.e. (1.2) fails) and it is known that both problemshave divergent RSPT. The Stark effect is more singular than the othertwo examples in that, as first noted by Oppenheimer [491], its boundstates turn into resonances, an issue we will discuss in Section 2.One might think, on the basis of these three examples that conver-gent RSPT is irrelevant to physics but that is wrong. First of all,the Kato–Rellich theorem implies that in the Born–Oppenheimer limit(i.e. infinite nuclear masses), the electronic energies are real analyticin the nuclear coordinates (at non-coincident points if the internuclearrepulsion is included); see [442, 475] for some of my work on Born–Oppenheimer curves.Moreover, we have the following interesting example on L ( R × R ) A ( β ) = − ∆ − ∆ − r − r + β | r − r | (1.6) where the Kato–Rellich theory applies. Up to a scale factor of Z − ,when β = Z − , this describes a two electron system moving around anucleus of charge Z . This A (0) is the sum of two independent hydrogenatoms so it has continuous spectrum [ − , ∞ ) and eigenvalues E n,m = − n − m ; n, m = 1 , . . . (1.7) WELVE TALES 5
For n or m equals 1, these are below − , so discrete and Theorem 1.1applies.There is a huge literature on the discrete eigenvalues of this system,especially the ground state. Some of it is summarized in [682, Example2.1]. I have a joint paper [299] on what happens at β c , the couplingwhere the ground state hits the continuous spectrum.The major theme of this section is that RSPT tells you somethingabout the eigenvalues, even when the series diverges. Before my work,the standard connection, where Kato [355] was the pioneer, concernedasymptotic series, a notion first formalized by Poincar´e [511] in 1886.Given a function, f , defined in a region R with 0 in its closure, we saythat f ( β ) has (cid:80) ∞ n =0 a n β n as an asymptotic series on R if an only if, forany N , we have thatlim | β |→ , β ∈ R ( f ( β ) − N (cid:88) n =0 a n β n ) β − N = 0 (1.8) Kato’s method allows one to prove that RSPT is asymptotic when R = (0 , B ) for any eigenvalue of the anharmonic oscillator, (1.3), andfor the Zeeman effect, (1.5), and the method in his book [363] allowsone to take R to be suitable sectors in the complex plane.(1.8) shows that f determines { a n } ∞ n =0 but since, for example, when R = (0 , B ), if (1.8) holds for f , it also holds for f ( β ) = f ( β ) +10 exp( − / β ), if we only know (1.8) and { a n } ∞ n =0 , we can’t sayanything about the value of f ( β ) for any particular fixed, non–zero β .Over the years, mathematicians have developed a number of methodsfor recovering a unique function among the several associated to a givenasymptotic series. Hardy [279] is a discussion of many of them. Twoof them – Pad´e and Borel summability are relevant to our discussionhere.Truncated Taylor series are polynomial approximations to a formalseries (cid:80) ∞ n =0 a n β n . Pad´e approximation involves rational approxima-tion (the name is after the thesis of Pad´e [493]; his advisor, Hermite,was a great expert on rational approximation). Given a formal series, (cid:80) ∞ n =0 a n β n , we say that f [ N,M ] ( β ) is the [ N, M ] Pad´e approximant if f [ N,M ] ( β ) = P [ N,M ] ( β ) Q [ N,M ] ( β ) ; deg P [ N,M ] = M, deg Q [ N,M ] = N (1.9) ? ? f [ N,M ] ( β ) − N + M (cid:88) n =0 a n β n = O (cid:0) β N + M +1 (cid:1) (1.10) ? ? B. SIMON
A formal power series is called a series of Stieltjes if it has the form a n = ( − n (cid:90) ∞ x n dµ ( x ) (1.11) for some positive measure, dµ on [0 , ∞ ) with all moments finite. Thisis related to the Stieltjes transform of µf ( β ) = (cid:90) ∞ dµ ( x )1 + xβ (1.12) since it easy to the that (cid:80) ∞ n =0 a n β n is an asymptotic series for such an f in any region of the form { β | | arg β | < π − (cid:15) } . A basic result onconvergence of Pad´e approximants is Theorem 1.2. (Stieltjes Convergence Theorem) If { a n } ∞ n =0 is a se- ? (cid:104) T1.2 (cid:105) ? ries of Stieltjes, then for each j ∈ Z , the diagonal Pad´e approximates, f [ N,N + j ] ( z ) , converge as N → ∞ for all β ∈ C \ [0 , ∞ ) to a function f j ( β ) given by (1.12) with µ replaced by µ j which obeys (1.11) (with µ = µ j ). The f j are either all equal or all different depending onwhether (1.11) has a unique solution, µ , or not. Remarks.
1. For proofs, see Baker [49, 50, 51] or Simon [680, Section7.7]2. Stieltjes [698] didn’t discuss Pad´e approximates by name butinstead studied continued fractions which lead to the result for j = 0 , | a n | ≤ AC n ( kn )! (1.13) for some k ≤ k = 1. One forms the Borel transform g ( w ) = ∞ (cid:88) n =0 a n n ! w n (1.14) which defines an analytic function in { w | | w | < C − } . Under the as-sumption that g has an analytic continuation to a neighborhood of[0 , ∞ ), one defines for β real and positive f ( β ) = (cid:90) ∞ e − a g ( aβ ) da (1.15) ? ? WELVE TALES 7 if the integral converges. Since (cid:82) ∞ e − a a n da = n !, formally, f ( β ) = (cid:80) ∞ n =0 a n β n . Here, one has a theorem of Watson [725]; see Hardy [279]for a proof: ? (cid:104) T1.3 (cid:105) ? Theorem 1.3.
Let Θ ∈ (cid:0) π , π (cid:1) and B > . Define Ω = { z | < | z | < B, | arg z | < Θ } (1.16) ? ? (cid:101) Ω = { z | < | z | < B, | arg z | < Θ − π } (1.17) ? ? Λ = { w | w (cid:54) = 0 , | arg w | < Θ − π } (1.18) ? ? Suppose that { a n } ∞ n =0 is given and that f is analytic in Ω and obeys (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( z ) − N (cid:88) n =0 a n z n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ AC N +1 ( N + 1)! (1.19) ? ? on Ω for all N . Define g ( w ) = ∞ (cid:88) n =0 a n n ! w n ; | w | < C − (1.20) ? ? Then g ( w ) has an analytic continuation to Λ and for all z ∈ (cid:101) Ω , wehave that f ( z ) = (cid:90) ∞ e − a g ( az ) da (1.21) ? ? My own work on the anharmonic oscillator was motivated by mythesis advisor, Arthur Wightman, who had the idea of exploring thisas a way of understanding QFT perturbation theory. He wanted toexploit an idea of Symanzik to use scaling. One looks at H ( α, β ) = − d dx + αx + βx (1.22) ? ? and notes that if λ is positive, then( U ( λ ) f ) x = λ / f ( λ / x ) (1.23) ? ? is unitary and U ( λ ) H ( α, β ) U ( λ ) − = λ − H ( αλ , βλ ) (1.24) So for any α, β, λ real with β, λ >
0, one has that E n ( α, β ) = λ − E n ( αλ , βλ ) (1.25) where E n is the n th eigenvalue of H ( α, β ).Wightman gave the problem to another graduate student, ArnieDicke, but they came to me with a technical problem they ran into.Then, in early 1968, I was a second year graduate student in physicsbut I had been charmed by Kato’s book [363] and was regarded as a B. SIMON local expert on some of the material. The problem was that U ( λ ) wasonly a bounded operator if λ > λ and they wanted (1.25) for complex λ .I came up with the following argument. In the region R = { α, β ∈ C , β / ∈ ( −∞ , } , H ( α, β ) is an analytic family of type A (I provedestimates like (1.2) for A = H ( α, β ) , β / ∈ ( −∞ ,
0] and B = x or x ).Thus, as long as the eigenvalue is simple at ( α , β ) ∈ R , we havethat E n ( α, β ) is analytic near ( α , β ). Since (1.25) holds for λ real,it holds for small complex λ by analyticity. (There was an issue ofeigenvalue labelling - there was no guarantee that if one went arounda loop starting and ending on R × (0 , ∞ ), that n couldn’t change.)Here, I missed a golden opportunity. I had proven an invariance ofdiscrete spectrum under complex scaling. It didn’t occur to me to askabout an operator like − d dx − βe − µx or − ∆ − r which like H ( α, β )has an analytic continuation for H ( λ ) = U ( λ ) HU ( λ ) − from real λ tocomplex λ . If I had I might have found Combes great discovery of ayear later (I’ll discuss his work in Section 2).After I found this, given that Dicke was bogged down in his con-struction of solution with the expected WKB asymptotics at infinity(which turned into his thesis and which he asked me to publish as anappendix to my long paper [581]), he and Wightman felt that I shouldexplore aspects of this problem beyond the existence of solutions thatDicke was looking at. I immediately noticed that (1.25) implies that E n (1 , β ) = β / E n ( β − / ,
1) (1.26) ? ? so since E n ( α,
1) is analytic near α = 0, E n (1 , β ) has a convergent seriesnear infinity, not in β − , but in β − / , so that E n (1 , β ) has a kind ofthree sheeted structure.In some of my work, I made an assumption that E n ( α,
1) has nonatural boundaries – this was proven to be true many years later(Eremenko–Gabrielov [177]) but for | arg α | < π/
3, as we’ll see shortly,it was proven there were no singularities at all in the same time frameas my paper.In 1968–69, Wightman was on leave in Europe and he thought aboutand talked to others about the anharmonic oscillator and wrote meletters. Andre Martin pointed out to him that the large β expansioncouldn’t converge for all β (cid:54) = 0. For, if it were, E n ( α,
1) would be anentire Herglotz function and so linear which one can easily see isn’ttrue for E n ( α,
1) = α / E n (1 , α − / ) shows thatlim α →∞ α ∈ R E n ( α, / √ α = E n (1 , WELVE TALES 9
I’d never seen the theorem about entire Herglotz functions which I’msure Martin got by using the Herglotz representation theorem. WhileI’d later often use that representation theorem heavily in my careerand even find a useful extension for meromorphic Herglotz functionson the disk [660], I’d never heard of it at the time. In those pre–Googledays, I couldn’t easily find much about Herglotz functions which wasgood because it forced me to find my own unconventional proof of theentire Herglotz theorem and that allowed me to prove that E n ( α, ω = e iθ , | θ | < π , one had that p + ωx + βx has a RS asymptotic perturbation series as β ↓ , β > E n (1 , β ) has an asymptotic series in { β | | arg β | < π/ − (cid:15), | β | < R (cid:15) } . This in turn implied that on thethree sheeted Riemann surface, there were an infinity of singularitieswith limit point 0 (on the natural three sheeted surface) and asymptoticphase ± π/ { a n } n =1 for the anharmonic oscillator ground state RS seriesand they did a numerical analysis leading to a conjecture of the large n asymptotics (I’ll say more about this subject in the next section).They also did a mathematically unjustified WKB analysis of the ana-lytic behavior of E n (1 , β ) which was consistent with what I had found.(I still remember that my first seminar outside Princeton was a physicstalk at Chicago where I made reference to the “notoriously unreliableWKB approximation”. Afterwards, a kindly older gentleman came upto me and introduced himself: “I’m the W of WKB”!).Early in 1969, I got a letter from Arthur Wightman that began “Thespecter of Pad´e is haunting Europe...”. Various theoretical physicistshad the idea of using diagonal Pad´e approximants on some field the-oretic Feynman series and Wightman suggested that I try it on theanharmonic oscillator. I’d never done any scientific computing (andhaven’t done any since!) but with the first 41 coefficients from theBender–Wu preprint and explicit determinantal formulae from Baker’sbook [50], it was straightforward.In those days, one did computer calculations by writing the programin Fortran on punch cards, submit the deck and waited a day to get back the results. My initial output was nonsense, but I realized I’d left out a( − n , fixed it, and the second time was golden! I computed f [ N.N ] ( β )for N = 1 , , . . . ,
20 and β = 0 . , . , . . . , f [ N,N ] ( β ) were monotone in N suggesting thatthe underlying series was a series of Stieltjes. I realized that withmy methods, to prove this, one needed to show that on { β | | arg β | <π } , the E j ( β ) have no natural boundaries and no eigenvalue crossing,equivalently the same for E j ( α,
1) within { α | | arg α | < π/ } . NickKhuri, a physicist at Rockefeller, heard of my work and invited me totalk while Martin was visiting there and I explained the situation tohim. Loeffel–Martin [450], using a clever argument tracking the zerosof eigenfunctions were able to show no eigenvalue crossing assumingone could make analytic continuation and I could show, using theirresults, that one could be sure one could analytically continue.The four of us (Loeffel, Martin, Simon and Wightman [451]) thenpublished an announcement putting everything together. The analyt-icity results implies that for a positive measure on (0 , ∞ ), one has that E j ( β ) = E − β (cid:90) ∞ dρ j ( x )1 + xβ (1.27) ? ? for all β ∈ C \ ( −∞ , k = 1, so the limits are the same and equal theeigenvalues.While this Pad´e is nice, the known scope where one can prove Pad´esummability is very limited. Loeffel et al. [451] note that their methodsimply that for m = 2 , , . . . , the RS series for the eigenvalues of p + x + βx m are series of Stieltjes so the diagonal Pad´e approximantsconverge. However, (1.13) holds for k = ( m −
1) so they only knewuniqueness when m = 2 ,
3. In fact, several years later, Graffi–Grecchi[252] proved that for the x oscillator, the f [ N,N + j ] converge as N → ∞ to j dependent limits, none of which is the eigenvalue!! Moreover, theLoeffel–Martin [450] method tracks zeros and so is limited to ODEsand there is no rigorous Pad´e result known for anharmonic oscillatorswith more than one degree of freedom.Borel summability turns out to be much more widely applicable.Shortly after the four author announcement appeared, I got contactedby Sandro Graffi and Vincenzo Grecchi whom I hadn’t previouslyknown. They enclosed a Xerox of the pages of Hardy’s book dealing WELVE TALES 11 with Watson’s Theorem and more importantly some numerical calcu-lations of the Borel sum of the x ground state (based on a not rigor-ously justified use of Pad´e approximants of the Borel transform, g , of(1.14)) which not only converged but more rapidly than ordinary Pad´eapproximants. I quickly determined that my techniques showed thehypotheses of Watson’s theorem held for x oscillators in any dimen-sion and that a higher order (i.e. kn ! instead of n !) Borel summabilityworks for the x m oscillator so we published a paper with these results[258]. Before leaving the issue of the perturbation series for the anhar-monic oscillator, I note that using the first 60 terms in the series andthe computer power available in 1978, Seznec and Zinn–Justin [565],using modified Borel summability and large order expansions, claimedto be able to find the ground state for all values of β to one part in10 !I wrote several papers on applying Borel summability in Φ cutofffield theory [582, 542] and other contexts [585, 586]. Avron–Herbst–Simon [32] proved Borel summability of Zeeman Hamiltonians andthere have been proofs by others of Borel summability of various quan-tum field theoretic perturbation series (Feynman diagram expansionof Schwinger functions): P ( φ ) [168], φ [455], Y [535], Y [456]. Aswe’ll see in the next section, there is a sense in which the Stark seriesis Borel summable.Before leaving asymptotic perturbation theory, I mention a strikingexample of Herbst–Simon [292] A ( β ) = − d dx + x − β x + 2 βx − βx If E ( β ) is the lowest eigenvalue, we proved that for all small, non–zero positive β < E ( β ) < C exp( − Dβ − )Thus E ( β ) has (cid:80) ∞ n =0 a n β n as asymptotic series where a n ≡
0. The as-ymptotic series converges but, since E is strictly positive, it convergesto the wrong answer!2. Complex Scaling Theory of Resonances (cid:104) s2 (cid:105) Our second tale also concerns eigenvalue perturbation theory, butin situations where the eigenvalue turns into a resonances. One ofthe simplest real physical examples where, at the time of my work,this was expected to occur involves the 1 /Z expansion of (1.6). Theeigenvalues E n,n of A (0) given by (1.7) when m, n ≥ E , = − > − . For β (cid:54) = 0,one expects the bound state to dissolve into a resonance. There is a standard physics textbook calculation called time–dependent perturbation theory (TDPT). The lifetime, τ , is by theWigner–Weisskopf formula τ = (cid:126) / Γ with Γ = 2Im E ( β ). The lead-ing order for Γ is called the Fermi golden rule and is given by Γ =Γ β + O( β ) whereΓ = ddλ (cid:104) Bϕ , ˜ P ( −∞ ,λ ) ( A ) Bϕ (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) λ = E (2.1) Here ˜ P is a spectral projection for A with the eigenvalue at E removedand A ϕ = E ϕ is the embedded eigenvalue with (cid:107) ϕ (cid:107) = 1. Usually,the right side of (2.1) is written as (cid:104) Bϕ , δ ( A − E ) Bϕ (cid:105) . This versionis from Simon [590]. There is a subtlety here that we won’t discuss indetail (but see [528] or [682, Example 3.2]): E , is actually a degener-ate eigenvalue and a subspace of the eigenspace at − has a differentsymmetry from the continuum at − , so only part of the eigenspacedissolves into resonances. These resonates are observed in nature andare called autoionizing states or Auger resonances.A second important model is the Stark Hamiltonian, (1.4). If β (cid:54) = 0,it is not hard to see that spec( A ( β )) is all of R (for A ( β )). For the oper-ator without Coulomb term, one can write down exp( itA ( β )) explicitly[27] and show that − /r doesn’t change the spectrum, indeed, waveoperators exist and are complete [65, 289]). Thus the discrete eigen-values are swamped by continuous spectrum. The theoretical physicsliterature based on a formal tunnelling calculation studied the leadingasymptotics of the width, which is O( β n ) for all n , and found that theleading order isΓ( β ) = (2 β ) − exp (cid:18) − β (cid:19) (cid:0) (cid:0) n (cid:1)(cid:1) (2.2) This formula, first found correctly by Lanczos [409], is called the Op-penheimer formula after [491].There were fundamental mathematical questions discussed byFriedrichs [195], who was the first person to look mathematically atissues of eigenvalues turning into resonances. First, in cases like theStark effect, where there are RSPT series but no eigenvalues, what isthe meaning of the perturbation coefficients. Second, what exactly isa resonance? Third, in a case like autoionizating states, what exactlyare the higher order terms of TDPT (the physics literature was unclearon this point) and is the series ever convergent?Before the complex scaling approach, there was the idea of solvingthe first problem by connecting the series to the asymptotics of thespectral projections of the perturbed operators. This notion, called
WELVE TALES 13 spectral concentration was pioneered by Titchmarsh [718] and Kato[352, 363] and later by others [109, 536]. It works well for the Starkeffect where the width is o( β n ) for all n so one can hope to provespectral concentration to all orders [529, 530] although it does notseem possible to fit a result like (2.2) into this framework. But for thecase of autoionizing states, the width go as cβ , c (cid:54) = 0 and there is onlyspectral concentration to first order.Howland wrote several papers [302, 303, 304, 305, 306] that addressedboth kind of models, but they required either the perturbation or someother object be finite rank so they didn’t cover either physical modelmentioned above.In two remarkable papers, Combes with his collaborators Aguilar andBalslev [5, 52] developed a framework to study the absence of singularspectrum that I realized was ideal to study autoionizing states. Theycalled it the theory of dilation analytic potentials but after the quantumchemists started using it in calculations, the name shifted to complexscaling.Consider first a two body potential, V ( x ) , x ∈ R ν and for θ ∈ R define ( U ( θ ) f )( x ) = e νθ/ f ( e θ x ) (2.3) ? ? which is a one parameter semigroup of unitary operators. Define, againfor θ ∈ R : V ( θ ) = U ( θ ) V U ( θ ) − (2.4) ? ? which is, of course, multiplication by V ( e θ x ). For general V , thisdoesn’t make sense for Im θ (cid:54) = 0, e.g. let V be a square well. Butfor some V ’s, one can analytically continue. Particular examples are V ( x ) = | x | − β , < β <
2, in particular for β = 1, where V ( θ ) continuesto an entire function and V ( x ) = e − µr /r where V ( θ ) can be continued(as a relatively bounded operator) so long as | Im θ | < π/ V is calleddilation analytic if V ( θ )( − ∆ + 1) − has an analytic continuation from R to all θ with | Im θ | < Θ for some Θ > H = − ∆ and H = H + V . Then H ( θ ) = U ( θ ) HU ( θ ) − (2.5) ? ? will have an analytic continuation to { θ | | Im θ | < Θ } as an analyticfamily of type (A).The Kato–Rellich theory is applicable. As in the last section, discreteeigenvalues are θ –independent at least for Im θ small. But since H ( θ ) = − e − θ ∆ + V ( θ ) (2.6) ? ? we know that if V ( − ∆ + 1) − is compact, then we have that H ( θ )has continuous spectrum e − Im θ [0 , ∞ ), i.e. the continuous spectrumrotates about the threshold 0.Balslev–Combes [52] analyzed the spectrum for N –body Hamilton-ian and found a spectrum like that shown in Figure 1 Figure 1.
Spectrum of a Complex Scaled HamiltonianInstead of continuous spectrum rotating about zero, it rotates abouteach scattering threshold. By an induction argument, one can provethat the set of thresholds is a closed countable set. An importantpoint is that as the spectrum swings down, it can uncover eigenvalueswhich then persist until perhaps hit by another piece of continuousspectrum when they can disappear. Combes and company interpretedthese complex eigenvalues as resonances.A key use Balslev–Combes made of their theory was to prove theabsence of singular continuous spectrum (see Section 6 below). One ofmy later results on complex scaling that I should mention is a quadraticform version [587] which has some significant technical simplifications,some of them involving work with Mike Reed on the spectrum of tensorproducts [523, 524, 589].In [590], I realized that complex scaling was an ideal tool for under-standing autoionizing states. One can prove that embedded eigenvaluesalso don’t move if Im θ is moved away from zero to positive values whilecontinuous spectrum does move. Thus in studying H + βW , one canlook at H ( θ ) + βW ( θ ). While E might be an embedded eigenvalue of H , so long as it is not at a scattering threshold of H , it is a discrete WELVE TALES 15 eigenvalue of H ( θ ) when θ = i(cid:15) with (cid:15) small and positive. So E willbecome a an eigenvalue, E ( β ), of H ( θ ) + βW ( θ ) given by a convergentpower series in β (if E is degenerate, there are extra subtleties). Ingeneral, Im E ( β ) <
0, i.e. embedded eigenvalues turn into resonances.The Rayleigh–Schr¨odinger series for E ( β ) provide an unambiguoushigher order TDPT which is convergent! Moreover, one can manipu-late the second order term to validate the Fermi golden rule and so geta rigorous proof of it.For the Stark effect, the conventional wisdom among mathematicalphysicists was that complex scaling can’t work. Because it was known(see, e.g. [289]) that − ∆ + F ˆ e · x for F (cid:54) = 0 and ˆ e a unit vector has noscattering thresholds, there was no place for the continuous spectrumof H ( θ ) = − e − θ ∆ + e θ F ˆ e · x (2.7) ? ? to go when Im θ is small and non–zero. But W. Reinhardt, a quantumchemist, was fearless and found [522] calculations gave sensible answers.This made I. Herbst reconsider the conventional wisdom [290]. Infact for Im θ ∈ (0 , π/ H ( θ ) defines a closed operator with emptyspectrum! So, since there is no place for the continuous spectrum to go,it disappears! It is, of course, known that a bounded operator cannothave empty spectrum (see, e.g. [680, Theorem 2.2.9]) but H ( θ ) is notbounded and H ( θ ) − has a single point, namely 0 in its spectrum; insome sense, H ( θ ) has ∞ as the only point in its spectrum. Herbst wasable to analyze [290] the Hydrogen Stark Hamiltonian whose resonanceenergies he showed have width that are O( F k ) for all k and had RSseries as asymptotic series.Herbst and I [293] then extended this work to analyze the StarkHamiltonian for general atoms. We also proved a kind of Borel summa-bility. The Rayleigh–Schr¨odinger perturbation series is Borel summableto a function defined about the positive imaginary axis in the F planewhose analytic continuation back to real F is the resonance. We provedthis for atoms. About the same time, Graffi–Greechi [253, 254, 256] dis-covered this for the hydrogen Stark effect using the separability of thatproblem into 1D problems (see below). Sigal [571, 572, 573, 574] andHerbst–Møller–Skibsted [291] have further studied Stark resonances inmulti–electron atoms proving that the widths are strictly positive andexponentially small in 1 /F .Harrell and I then wrote a paper [280] that was able to analyzethe small coupling behavior of the imaginary part of some resonanceenergies that are exponentially small. Essentially, this allowed a rig-orous proof of some results obtained earlier by theoretical physicists using a formal WKB analysis. First of all we proved the Lanczos–Oppenheimer formula (2.2). As noted earlier by Herbst–Simon [292]this implies asymptotics of the perturbation coefficients E ( F ) ∼ ∞ (cid:88) n =0 A n F n (2.8) ? ? A n = − n +1 (2 π ) − (2 n )! (cid:18) (cid:18) n (cid:19)(cid:19) (2.9) ? ? since one can write f ( x ) = (cid:72) γ (2 πi ) − f ( z ) z − x dz where γ is a contour that isa small circle with a loop around the negative axis (in the − F variable)and a large circle.In the context of the anharmonic oscillator, the same idea of preciseasymptotics of RS coefficients occurred earlier than the work of Herbst–Simon and Harrell–Simon. As noted in Section 1, Bender–Wu [60] hadcomputed the first 75 coefficients for the ground state E (1 , β ) and theydid a numerical fit and conjectured that a n = 4 π − / ( − n +1 (cid:0) (cid:1) n +1 / Γ( n + ) (cid:0) (cid:0) n (cid:1)(cid:1) (2.10) They had the leading constant to 8 decimal places and guessed itsanalytic form. In my anharmonic oscillator paper [581], I noted that(2.10) was equivalent to leading asymptoticsIm E (1 , β ) ∼ β = − b + i π − / b − / e − / b (2.11) Without noticing my remark, Bender and Wu noted [61] that (2.11),and so (2.10), follow from a formal WKB calculation of the tunnellingin a potential x − bx . Harrell-Simon [280] have rigorous proofs of(2.11) and so (2.10). Hellfer-Sj¨ostrand [284] proved Bender-Wu typeformulae for higher dimensional oscillators.Harrell–Simon uses ODE (i.e. 1D) techniques. The Stark effectcan be separated into 1D problems in elliptic coordinates (noted byJacobi [325] in classical mechanics and then Schwarzschild [555] andEpstein [175] in old quantum theory and in parabolic coordinates bySchr¨odinger [554] and Epstein [176]) and this was later used mathemat-ically Titchmarsh [718, 719], Harrell–Simon [280] and by Graffi–Grecchiand collaborators [253, 254, 255, 257, 259, 59, 83, 256].The Zeeman effect for Hydrogen can be reduced to a two dimen-sional problem. Avron [26] used this and an instanton calculation oftunnelling (see section 8) to formally compute the asymptotics for RS WELVE TALES 17 coefficients for the ground state of the Zeeman Hamiltonian (1.5). E k = (cid:18) π (cid:19) / ( − k +1 π − k Γ (cid:18) k + 32 (cid:19) (cid:18) (cid:18) k (cid:19)(cid:19) (2.12) ? ? Hellfer-Sj¨ostrand [284] then gave a rigorous proof of this using PDEtechniques.Quantum Chemists embraced the complex scaling method to do cal-culations of resonance energies in atoms and molecules. I wrote a reviewof the mathematical theory [613] for a joint conference. The calcula-tions for molecular resonance curves were done in a Born–Oppenheimerapproximation with fixed nuclei which lead to potentials which are scaleanalytic. I introduced exterior complex scaling Simon [615] to justifywhat they did. A more elegant approach (smooth exterior scaling) wassubsequently developed by Hunziker [317] and G´erard [215].I should note that I have reason to believe that, at least at one time,Kato had severe doubts about the physical relevance of the complexscaling approach to resonances. [280] was rejected by the Annals ofMathematics, the first journal it was submitted to. The editor told methat the world’s recognized greatest expert on perturbation theory hadrecommended rejection so he had no choice. I had some of the reportquoted to me. The referee said that the complex scaling definitionof resonance was arbitrary and physically unmotivated with limitedsignificance. My review of Kato’s work on non-relativistic quantummechanics (henceforth NRQM) [682, Part 1, pg. 154-155] has a longdiscussion of why I believe the complex scaling definition is physicallyrelevant with many references to the literature.I should mention that I used complex scaling [594] to show N –bodysystems with local potentials that can be continued to the right halfplane (in particular, with Coulomb potentials) can’t have positive en-ergy bound states or thresholds.While I’ve focused on the complex scaling approach to resonances,there are other methods. One, called distortion analyticity, workssometimes for potentials which are the sum of a dilation analyticpotential and a potential with exponential decay (but not neces-sarily any x –space analyticity). The basic papers include Jensen[333], Sigal [570], Cycon [116], and Nakamura [480, 481]. Someapproaches for non-analytic potentials include G´erard-Sigal [216],Cattaneo–Graf–Hunziker [97], Cancelier–Martinez–Ramond [85] andMartinez–Ramond–Sj¨ostrand [463]. There is an enormous literatureon the theory of resonances from many points of view. I should men-tion a beautiful set of ideas about counting asymptotics of resonancesstarting with Zworski [743]; see Sj¨ostrand [692] for unpublished lectures that include lots of references, a recent review of Zworski [744] and thebook of Dyatlov–Zworski [162] (I have one paper related to these ideas[659]). The form of the Fermi Golden Rule at Thresholds is discussedin Jensen–Nenciu [335]. A review of the occurrence of resonances in NRQuantum Electrodynamics and of the smooth Feshbach–Schur map isSigal [575] and a book on techniques relevant to some approaches toresonances is Martinez [462].Finally, I note that these two sections have dealt with eigenvalueperturbation theory. I’ll return in Section 6 to a different issue involvingperturbations that give birth to eigenvalues from the edge of continuousspectrum and in Section 8 to eigenvalues at limiting values of couplingconstant, namely − (cid:126) ∆ + V ( x ) as (cid:126) ↓ Statistical Mechanical Methods in EQFT (cid:104) s3 (cid:105) The fifteen years following 1965 saw the development of a subjectknown as constructive quantum field theory (CQFT) which success-fully constructed interacting quantum fields in 2 and 3 space-time di-mensions obeying all the Wightman axioms [734, 701, 346]. Because ofthe failure to get to 4 space-time dimensions (except for some negativeresults [7, 198]), the long lasting impact to rigorous quantum physicshas been more limited than initially hoped although extending to thephysically relevant 4 dimensional case is a million dollar problem [327].Still, the spinoff to various areas of mathematics and theoretical physicshas been substantial.My main goal in this section is to focus on my work, much of it jointlywith Francesco Guerra and Lon Rosen, on using methods from classicalstatistical physics to study Bose CQFT, but I’ll begin with some ofmy other work motivated by CQFT that had important mathematicalspinoffs.CQFT was developed by many researchers including F. Guerra, K.Osterwalder, L. Rosen, R. Schrader, I. Segal, E. Seiler, T. Spencer, A.Wightman and especially J. Glimm, A. Jaffe and E. Nelson. I refer thereader to the books of Simon [595] and Glimm-Jaffe [239].The initial work mainly on ( ϕ ) theories focused on the Hamilton-ian viewpoint where controlling spatially cutoff theories is hard becausethe operators act on an infinite number of variables and the potentialis not bounded from below (we use ( X ) d shorthand to describe theorieswhere d is the number of space-time dimensions and X an abbreviationfor the interaction term). The first breakthrough was by Nelson [482]who realized that the free Bose Hamiltonian, H , in a periodic box inone space dimension, viewed as an infinite sum of harmonic oscillators WELVE TALES 19 (with different frequencies), could be realized as a Gaussian process byshifting from dx to ϕ dx , where ϕ is the ground state, so that H actedon R ∞ with a Gaussian probability measure, dµ . The operator H wasthen realized as a pure Dirichlet form (i.e. (cid:104) ψ, H ψ (cid:105) = (cid:82) |∇ ψ | dµ ).For differential operators, this shift to ground state measure and Dirich-let form goes back to Jacobi (!) and since Nelson’s work has been usedmany times in mathematical analysis of quantum theories, e.g. [193].In this representation, Nelson proved that (cid:107) e − tH f (cid:107) p ≤ (cid:107) f (cid:107) p (3.1) for all f ∈ L p ( R ∞ ) and all t > T > (cid:107) e − T H f (cid:107) ≤ C (cid:107) f (cid:107) (3.2) for some fixed C and all f ∈ L . He also showed that while the ( ϕ ) spatially cutoff interaction, V , is not bounded from below, it obeys (cid:90) e − sV dµ < ∞ all s > V ∈ (cid:92) p< ∞ L p ( R ∞ , dµ ) (3.4) and most importantly that (3.1)-(3.4) imply that H + V is boundedfrom below.Two important followups were by Glimm [235], who proved that(3.2) plus a mass gap imply that by increasing T , (3.2) holds with C = 1 (this allows removing the need for Nelson to restrict to periodicboundary conditions) and by Federbush [181], who used interpolationto prove that (cid:107) e − sH f (cid:107) p s ≤ C s (cid:107) f (cid:107) with p s ↓ s ↓ H + V on D ( H ) ∩ D ( V ) for spatially cutoff ϕ . This was accomplished by Glimm-Jaffe [236] who proved it using additional estimates beyond those ofNelson and subsequently by Segal [558, 559, 560] who only needed theestimates (3.1)-(3.4).At this point, my work enters via a widely quoted joint paper withHøegh-Krohn [684] entitled Hypercontractive semi-groups and two di-mensional self-coupled Bose fields . We abstracted and simplified Se-gal’s self-adjointness result. One significant aspect was inventing theterm “hypercontractive” for groups obeying (3.1) and (3.2) (Nelsoncomplained to me that since (3.2) has a C which might not be one,we should have used “hyperbounded” but I replied that hypercon-tractive sounded better). Other terms that I’ve introduced that have caught on include Agmon metric, almost Mathieu equation, Berry’sphase, Birman-Schwinger bound, CLR inequality, CMV matrix, cou-pling constant threshold, diamagnetic inequalities, HVZ theorem, Katoclass, Kato’s inequality, ten martini problem, Verblunsky coefficientsand ultracontractivity.Hypercontractivity and its differential version, logarithmic Sobolevinequalities (first completely explicated by Gross [267]), have had anenormous number of applications outside quantum field theory; theyare even used in Perelman’s proof of the Poincar´e conjecture. See [679,Section 6.6] for a discussion of the various sides of the mathematicaltheory with historical notes, additional references and presentation ofsome of the applications. Several years later, in 1983, Brian Davies andI [127] found a variant of hypercontractivity called ultracontractivitywhich has evoked considerable mathematics.Before turning to the discussion of statistical mechanical methodsin QFT, I should mention another aspect of my work in CQFT withmathematical spinoff. I wrote a series of papers with E. Seiler [561, 562,563, 564] on the Yukawa QFT in two space-time dimensions, aka Y ,that developed some mathematical tools in the theory of trace idealsthat have had many applications including to quantum informationtheory.The work on statistical mechanical methods depends on the secondbig breakthrough in CQFT, namely Euclidean Quantum Field The-ory (EQFT). The Wightman axioms show that the Wightman func-tions (vacuum expectation values of the product of quantum fieldsas tempered distributions on Minkowski space) of any QFT can beanalytically continued in time to pure imaginary time differences andthat these continued functions are invariant under the Euclidean group.Schwinger [556] first emphasized this, so the analytic continuation toimaginary times are sometimes called Schwinger functions. Symanzik[704, 705] noted the analogy between classical statistical mechanics andEQFT focusing on the analog of the Kirkwood-Salzburg equations.The central development was due to Nelson [484, 485]. He under-stood that for Bose QFT, EQFT is essentially an infinite dimensionalpath integral with the extra bonus of Euclidean invariance. A keyrole was played by the extension of the Feynman-Kac formula thatGuerra-Rosen-Simon called the Feynman-Kac-Nelson formula. Thisimmediately implied a symmetry, later dubbed Nelson’s symmetry: (cid:104) Ω , e − tH (cid:96) Ω (cid:105) = (cid:104) Ω , e − (cid:96)H t Ω (cid:105) (3.5) where H (cid:96) = H + (cid:82) (cid:96) : P ( ϕ ) : ( x ) dx is the spatially cutoff Hamiltonianand H Ω = 0. Nelson also realized the key multidimensional Markov WELVE TALES 21 property which allowed one to go from Euclidean fields to Minkowskifields (later Osterwalder-Schrader [492] found an alternate way to dothis, which, because it extended to Fermi fields and provided necessaryand sufficient conditions, supplanted this part of Nelson’s approach).Nelson gave a few lectures on this new approach in Princeton early in1971 and lectured at a Berkeley summer school that summer attendedby many experts on CQFT. Even though this work eventually rapidlyrevolutionized the subject, initially, it had little impact. I think part ofthe reason this happened was that the language, especially as presentedby Nelson, was so foreign to the functional analysts working in the field,part was that Nelson’s lectures seemed obscure and, most importantly,his original work provided no new technical results in conventionalCQFT. Indeed, the only CQFT technical result was a new proof of alower bound for ( ϕ ) theories E (cid:96) ≡ inf spec( H (cid:96) ) ≥ − c(cid:96) − d (3.6) a result originally proven by Glimm-Jaffe [237]. Nelson’s proof wasmuch simpler than theirs but its impact was lessened by a simple proofthat I found (Simon [588]) shortly before Nelson.The work that made Nelson’s theory take off was a remarkable noteof Francesco Guerra [269], then a postdoctoral visitor at Princeton.Guerra was out of town when Nelson lectured but he got notes fromSergio Albeverio, then another fellow postdoc. Guerra realized that(3.5) and E (cid:96) = lim t →∞ − t log (cid:104) Ω , e − tH (cid:96) Ω (cid:105) (3.7) ? ? provided tools to study E (cid:96) and Ω (cid:96) , the vector with H (cid:96) Ω (cid:96) = E (cid:96) Ω (cid:96) (for ex-ample, these two equations immediately imply that (cid:96) (cid:55)→ E (cid:96) is concave).He proved that E (cid:96) /(cid:96) had a limit, α ∞ , and that |(cid:104) Ω (cid:96) , Ω (cid:105)| = O( (cid:96) − k ) forall k . This was way beyond anything that obtained via the purelyoperator theory used previously in CQFT.Indeed, I have a vivid memory of how I first learned of these results.Guerra had been visiting Princeton at that point for about 18 months.He was very quiet - I’d probably exchanged only a few words withhim and he’d given no talks. Wightman told me that Guerra hadasked Wightman to set up a meeting with Lon Rosen (another postdocand a student of Glimm with several significant CQFT results) andme and we met in Wightman’s office in early January, 1972. Guerrabegan by writing three facts that he was going to prove. Lon and I latercompared notes and we had the same thought “yeah, sure, you’re goingto do that”. These went so far beyond what was known that it wasliterally unbelievable. He began by writing (3.5) on the blackboard which we’d seen since it was part of Nelson proof of (3.6) and tenminutes later, he’d proven the three facts. We were shell shocked!After Guerra told us of these results, Lon, Francesco and I beganworking together on exploiting these ideas (our work went through twophases - first we mainly exploited consequences of (3.5) and similarresults but later we fully embraced the Euclidean viewpoint). In shortorder, we found [271, 272] improvements on what Guerra had found:first E (cid:96) = − α ∞ (cid:96) − β ∞ + o(1) as (cid:96) → ∞ and secondly, for some c, d > (cid:96) ≥
1, one has that e − c(cid:96) ≤ |(cid:104) Ω (cid:96) , Ω (cid:105)| ≤ e − d(cid:96) (see Lenard-Newman[427] for further developments on these subjects). Moreover, we found anew and much simpler proof of some bounds of Glimm-Jaffe [238] thatallow one to show that limit points of the cutoff Wightman functions(as the spatial cutoff in H (cid:96) is removed) are tempered distributions.The above mentioned work of Guerra and GRS got the attentionof experts in CQFT and virtually all papers in the subject after early1972 used the EQFT framework. I recall that a few weeks after GRSstarted working together, Glimm came to Princeton to talk about thebounds in [238] and spent the hour seminar sketching their subtle proof.Afterwards, Francesco, Lon and I waylaid him and explained in 10 min-utes the short proof we had found using a extended version of Nelson’ssymmetry. It was Glimm’s chance to be shell shocked!The further introduction of techniques from rigorous statistical me-chanics and, in particular, the use of correlation inequalities, the ma-jor accomplishment highlighted in this section, were introduced in twopapers, one by Guerra-Rosen-Simon [274] and one by Griffiths-Simon[266]. GRS [274] was a long paper, so long that the Annals of Mathe-matics broke it into two parts so it spread it between two issues. Amongother things, it provided a detailed exposition of EQFT so that it andmy book on the P ( ϕ ) theory (Simon [595], based at lectures I gave atthe ETH) served as the standard references on the subject for a time.The most important set of ideas in GRS [274] involve the latticeapproximation. Our work was announced [273] a year earlier thanWilson’s work [736] on lattice QCD, which of course went much furtherby allowing Fermion and Gauge fields albeit without mathematicalrigor. (It appears from Wilson’s historical note [737] that he didn’tstart to think about lattice approximations to EQFT until early 1974while we were already working on it in the spring of 1972; that said,there is no reason to think that Wilson knew of our work in 1974 or evenin 2005!). The free EQFT is a Gaussian random field with covariance( − ∆ + m ) − . One gets the lattice approximation by replacing − ∆by a finite difference operator. Since it is the inverse of the covariancethat appears in the exponent of the Gaussian field, the free lattice field WELVE TALES 23 is formally Z − exp − (cid:88) | i − j | =1 ( s i − s j ) (cid:89) j ∈ Z e − m s j ds j (3.8) which is an Ising type ferromagnet with nearest neighbor interactionsand spins lying in R (rather than just ± e − as ds . While (3.8) is a formal infinite product, if one puts it in afinite box, the spins lie in R k and the product is a simple finite measure.An analysis of the interaction just changes e − as to e − Q ( s ) for a suitablesemibounded even polynomials.A powerful tool in the statistical mechanics of spin systems is cor-relation inequalities, a method initiated by Griffiths [261, 262, 263]whose inequalities were extended by Kelly-Sherman [371] (hence GKSinequalities). A different set of inequalities are due to Fortuin, Kaste-lyn and Ginibre [188] (hence FKG inequality). Relevant to EQFT areversions tailor made for spins with continuous values due to Ginibre[234] for GKS and Cartier [94] for FKG. With these extensions, GRS[274] obtained GKS and FKG inequalities for Euclidean P ( ϕ ) theories.The most important application of these correlation inequalities(namely of GKS) is to show monotonicity in volume of the so-calledhalf-Dirichlet Schwinger functions, a suggestion of Nelson [486], ex-ploited by GRS [274] to obtain P ( ϕ ) quantum fields obeying all theWightman axioms except perhaps uniqueness of the vacuum (for thislast axiom, see below). It should be mentioned that the earliest con-struction of P ( ϕ ) theories (indeed the first construction of non-trivialexamples of theories obeying all the Wightman axioms, albeit in 2space-time dimensions), using cluster expansions, was by Glimm-Jaffe-Spencer [240] (using cluster expansions) for λP ( ϕ ) theories with small λ and, then, by Spencer [697] for P ( ϕ ) + µϕ with | µ | large, The Nelson-GRS work (for P ( X ) = Q ( X ) + µX with Q even) was the first resultswithout restrictions on coupling constant.The second application we mention were results by Simon [591] who,following work of Lebowitz [423] on spin systems, used the FKG in-equalities to show that decay of the truncated two point function dom-inates the decay of all the truncated vacuum expectation values. Thismeans to prove uniqueness of the vacuum (respectively, existence ofa mass gap), it is enough to prove that as x − y → ∞ , one has that (cid:104) ϕ ( x ) ϕ ( y ) (cid:105) − (cid:104) ϕ ( x ) (cid:105)(cid:104) ϕ ( y ) (cid:105) goes to zero (respectively goes to zero ex-ponentially).While the work of Ginibre and Cartier nicely prove GKS and FKGinequalities for fairly general single spin distributions, there are other results for ± ± s and s and let t = ( s + s ). Then t takes values 0 , ± s and s are uncoupled, the weights are , , rather than equal weights. If we find a coupling with energy H so thatthe Gibbs weight e − H has the values 2 , ,
2, then the adjusted wights areall equal. We thus pick H = − (log 2) t which is ferromagnetic. Thusany correlation inequality that holds for ferromagnetically coupled spin1 / − as − bs ) ds ( a > , b ∈ R ) as a limit of scaled spin 1 / P ( ϕ ) the-ories with P ( X ) = aX + bX + µX, a, µ ≥ , b ∈ R . This in turn canbe used prove that when µ > P is of (even) degree larger than 4 thenexp( − P ( x ) dx ) dx cannot be approximated by ferromagnetic arrays of ± ϕ theories.I should remark that correlation inequalities are useful in the studyof Schr¨odinger operators on R ν . For example, it is known [600, 170],using GHS inequalities, that if V ( x ) is an even function on R with V (cid:48)(cid:48)(cid:48) ( x ) ≥ x > E < E < E are the first three eigenvaluesof − d /dx + V ( x ), then E − E ≥ E − E . And, in Section 7, we’lldiscuss applications of FKG inequalities to Schr¨odinger operators inmagnetic fields.While I only worked on CQFT in two space-time dimensions, thereis some deep work by others on the three dimensional case. This, aswell as work by others on two dimensions, is presented in the book byGlimm-Jaffe [239].I should close the discussion of my work in CQFT by mentioning apaper with Fr¨ohlich [202] that, among other things, constructs P ( ϕ ) theories obeying all the Wightman axioms for any semibounded poly-nomial P . It relies on Spencer’s large µ expansion [697] and FKGinequalities. 4. Thomas–Fermi Theory ? (cid:104) s4 (cid:105) ? In 1972-73, Elliott Lieb and I found results on the Thomas-Fermi(TF) theory that we announced in 1973 [440] with full details onlypublished in 1977 [441] due, in part, to a long journal backlog. We
WELVE TALES 25 first of all established existence and uniqueness of solutions to the TFequations for neutral (and positive ionic) atoms and molecules and,more importantly, proved that TF theory was an exact approximationto quantum theory in suitable Z → ∞ limits. Since then, an entireindustry has been spawned from this work.TF theory goes back to Thomas [714] and Fermi [184] in 1927 nearthe dawn of quantum mechanics as an approximation expected to bevalid in regions of high electron density. Interestingly enough, it ap-proximated a linear equation in 3 N variables as N → ∞ by a non-linearequation in 3 variables! They originally found their basic equation us-ing a Fermi surface heuristic argument but we relied on the 1932 ap-proach of Lenz [429] who used energy functionals giving birth to thedensity functional method of atomic and molecular physics that hasbecome such a standard that the 1998 Nobel Prize in Chemistry wasawarded to Walter Kohn “for his development of the density-functionaltheory”.In units where (cid:126) (cid:0) (cid:1) / (2 m ) − = 1 (4.1) (where m is the electron mass and we assume the electron has 2 spinstates - state counting is important because one assumes Fermi statis-tics), the Lenz functional is E ( ρ ; V ) = 35 (cid:90) ρ / ( x ) d x + 12 (cid:90) ρ ( x ) ρ ( y ) | x − y | d xd y − (cid:90) ρ ( x ) V ( x ) d x (4.2) Here ρ ( x ) is the electron density, so, if there are N electrons, we havethat (cid:90) ρ ( x ) d x = N (4.3) and V ( x ) is the one electron potential; for a molecule with nuclearcharges z , . . . , z k at distinct points R , . . . , R k , we have that V ( x ) = k (cid:88) j =1 z j | x − R j | (4.4) We set Z = k (cid:88) j =1 z j (4.5) The last term in (4.2) is just the interaction of the electrons withthe nuclei and is exact, not an approximation. The second term is an electron repulsion and assumes no electron correlation so that the twopoint density is ρ ( x, y ) = ρ ( x ) ρ ( y ) (4.6) ? ? which cannot be even approximately true unless N is large, since (cid:82) ρ ( x, y ) d xd y = N ( N − (cid:82) ρ ( x ) ρ ( y ) d xd y = N .The first term relies on a quasi-classical calculation. If one has N par-ticles in a box, Ω, of size | Ω | with ρ = N/ | Ω | and fills phase space byputting particles in { p | | p | ≤ p F } , then N = π p F | Ω | / h (2 in thedenominator from 2 spin states) by the rule that each particle takesvolume h in phase space. The total energy of this is then C | Ω | ρ / with an explicit C which explains where the first term in (4.2) comesfrom. The choice (4.1) leads to C = 3 /
5. Of course, the notion ofstates taking h in phase space is an approximation justified in a large N limit by Weyl’s celebrated eigenvalue counting result (see later and[680, Section 7.5] for exposition and references).The Euler-Lagrange equation with Lagrange multiplier to take thecondition (4.3) into account with the restriction ρ ( x ) ≥ ϕ ≥ ϕ ( x ) = V ( x ) − (cid:90) ρ ( x ) | x − y | d y (4.7) ? ? ρ / ( x ) = (cid:26) ϕ ( x ) − ϕ , if ϕ ( x ) ≥ ϕ , if ϕ ( x ) ≤ ϕ (4.8) ? ? This is the Thomas-Fermi integral equation which implies the ThomasFermi PDE ∆ ϕ = [max( ϕ − ϕ , / (4.9) One result that Lieb and I proved [440, 441] is the following: ? (cid:104) T4.1 (cid:105) ? Theorem 4.1.
Let V be given by (4.4) and N, Z given by (4.3) / (4.5) .Then E ( ρ ; V ) is well defined if ρ ≥ lies in L ∩ L / . Moreover:(a) If N ≤ Z , there is a unique minimizer of E ( ρ ; V ) among those ρ ’s obeying (4.3) .(b) If N > Z , there is no minimizer of E ( ρ ; V ) among ρ ’s obeying (4.3) (c) If N < Z , the minimizing ρ has compact support and obeys theTF integral equation for some ϕ > and is real analytic on the openset { x | ϕ ( x ) > ϕ ; x / ∈ { R j } kj =1 } .(d) If N = Z , the minimizer minimizes E ( ρ ; V ) on all ρ ∈ L ∩ L / , ρ ≥ without any condition (4.3) . This minimizing ρ obeys theTF integral equation with ϕ = 0 . One has that, for all x , ϕ ( x ) > WELVE TALES 27 and so ρ ( x ) > on all of R . ϕ is real analytic on R \ { R j } kj =1 and ρ ( x ) ∼ | x | − (4.10) ? ? as x → ∞ . Remarks.
1. The only prior results on existence were for the neutralatomic case ( k = 1 , R = 0 , ϕ = 0) where one looks for sphericallysymmetric solutions of (4.9). Since, if ϕ is spherically symmetric, onehas that ∆ ϕ = r − ( rϕ ) (cid:48)(cid:48) , we see that if Y ( r ) = rϕ ( rω ), then (4.9) when r (cid:54) = 0 is equivalent to Y (cid:48)(cid:48) ( r ) = r − / Y ( r ) / (4.11) which goes back to the work of Thomas and Fermi. Thomas noticedthat Y ( x ) = 144 x − solves (4.11) (which leads to ϕ = 144 x − and ρ = ϕ / = 1728 | x | − ). In 1929, already, Mambriani [457] provedexistence and uniqueness of solutions of (4.11) with lim r ↓ Y ( x ) = a and lim x →∞ Y ( x ) = 0; see Hille [296] for further work. But Lieb-Simon had the first results on existence and uniqueness going beyondthe spherically symmetric case. We note that uniqueness of sphericallysymmetric solutions of the PDE doesn’t prove that the minimizer for E is spherically symmetric nor that the minimizer is unique.2. Sommerfeld [695] suggested that the singular solution 144 x − should control general asymptotics of the TF PDE and that was provenby Hille [296] in the spherically symmetric case and by Lieb-Simon[441] in the non-central case. We used subharmonic comparison ideas,a technique we learned from Teller [713], who used it in a differentcontext.3. Uniqueness of minima follows from strict convexity of E , i.e.0 < θ < , ρ (cid:54) = ρ ⇒ E ( θρ + (1 − θ ) ρ ) < θ E ( ρ ) + (1 − θ ) E ( ρ )since one term in E is linear and the other two are strictly convex.4. Existence uses what has come to be called the direct method ofthe calculus of variations (see, for example Dacorogna [118]). Namely,one looks at { ρ | ρ ∈ L ∩ L / , ρ > , (cid:82) ρdx ≤ N } which is compact ina suitable weak topology (if ≤ N is replaced by = N , it is not weaklyclosed, so not compact) and one proves that ρ (cid:55)→ E ( ρ ) is weakly lowersemicontinuous. A potential theory argument shows that if N ≤ Z , theminimizer has (cid:82) ρ = N but if N > Z , the minimizer obeys (cid:82) ρ = Z .These weak compactness, lsc ideas are now standard analysis but, atthe time, while they were used in some areas, they were not widelyknown in mathematical physics. While Lieb eventually became a worldexpert in subtle extensions of this method, he learned the necessaryfunctional analysis from me at the time of our work. With the existence out of the way, we turned to figuring out theconnection to atomic physics. In this regard, there are several reasonsthat the TF theory might not have anything to do with true quantumsystems. As we saw, in the neutral case, ρ T F decays as x → ∞ as C | x | − but true atomic bound states decay exponentially (see Section 6below; O’Connor’s work was done before Lieb and I were working). Aswe’ll see, scaling shows that in the atomic case the TF density shrinks as Z grows (at a Z − / rate), while true atoms expand in extent (althoughit might be that atomic radii, defined as where Z − Z → ∞ , they certainly don’t shrink). Finally,it is a result of Teller [713] that molecules don’t bind in TF theorywhile they do exist in nature. For technical reasons, Teller had a shortdistance cutoff in the Coulomb potential in his argument leading somepeople to question whether his result held in TF theory without cutoffs,but, it does, as Lieb and I showed. Interestingly enough, this apparentnegative result in TF theory was a key, several years later, in the elegantLieb-Thirring proof [446] of the stability of matter!Early in our work, Lieb understood why none of these issues wereproblems in connecting TF theory to quantum mechanical atoms. TFtheory describes the cores of atoms while chemistry involves the out-ermost electrons so it isn’t surprising that molecules don’t bind in TFtheory - it is an expression of the repulsion of the cores. The | x | − Sommerfeld asymptotics describes the mantle of the core while expo-nential decay describes the last few electrons. I still remember the startof our collaboration while we were both visitors at IHES in the fall of1972. Lieb had the idea that Weyl type estimates should show thatTF was a proper semiclassical limit of atoms. At the end of a long dayof discussing this idea, I told him of Teller’s result which I’d learnedabout in a course taught by Wightman, so since TF theory didn’t bindatoms, it couldn’t describe physics. The next morning Lieb walked inand said to me: “Mr. Dalton’s hooks are in the outer shell.” In otherwords, chemistry had nothing to with region in which a leading quasi-classical limit is valid. (The notion behind density functional theory isthat chemistry can be connected to non-leading terms).One key to the large Z results is scaling. The following is easy tocheck. If Z > V Z ( x ) = Z / V ( Z / x ); ρ Z ( x ) = Z ρ ( Z / x ) (4.12) ? ? then E ( ρ Z ; V Z ) = Z / E ( ρ ; V ); (cid:90) ρ Z ( x ) d x = Z (cid:90) ρ ( x ) d x (4.13) ? ? WELVE TALES 29
In particular if E V ( Z ; N ) = inf {E ( ρ ; V Z ) | ρ ∈ L ∩ L / , ρ ≥ (cid:90) ρ ( x ) d x = N } (4.14) ? ? then E V ( Z ; nZ ) = Z / E V (1; n ) (4.15) Given z , . . . , z k , R , . . . , R k , we let E T F ( N ; z , . . . , z k ; R , . . . , R k ) bethe TF energy (i.e. minimum of E ( ρ ; V ) with V given by (4.4) over ρ ’sobeying (4.3)). Then (4.15) says that E T F ( nZ ; z Z, . . . , z k Z ; Z − / R , . . . , Z − / R k )= Z / E T F ( n ; z , . . . , z k ; R , . . . , R k ) (4.16) ? ? We next describe quantum atomic energies. Let H phys be those ele-ments in L ( R N ; C N ) which are functions of N points x , . . . , x N in R and spins σ , . . . , σ N in C which are antisymmetric under permu-tations of ( x j , σ j ) (see [680, Section 7.9] for more on this formalism).On H phys let H = − N (cid:88) j =1 (cid:126) m ∆ j + (cid:88) i For any distinct R , . . . , R k and positive z , . . . , z k , and n > , we have that lim Z →∞ Z − / E Q ( nZ ; z Z, . . . , z k Z ; Z − / R , . . . , Z − / R k )= E T F ( n ; z , . . . , z k ; R , . . . , R k ) (4.20) Moreover, if n ≤ (cid:80) kj =1 z j , then lim Z →∞ nZ − ρ Q ( Z − / x ; z Z, . . . , z k Z ; Z − / R , . . . , Z − / R k )= ρ T F ( x ; n ; z , . . . , z k ; R , . . . , R k ) (4.21) in the sense of convergence integrated over x in any fixed open set. Our proof of (4.20) uses the method of Dirichlet-Neumann bracket-ing. This goes back to Weyl [732] as formalized by Courant-Hilbert[111] (see [13] for the discrete analog and [680, Section 7.5] for anothertextbook discussion). They used it to count eigenvalues of the Lapla-cian in regions with smooth boundary. It was later used to prove thatwhen V ∈ C ∞ ( R ν ), then as λ → ∞ , one has that N ( λV ), the numberof negative eigenvalues of − ∆ + λV on L ( R ν ) obeyslim λ →∞ λ − ν/ N ( λV ) = (2 π ) − ν τ ν (cid:90) max( − V ( x ) , ν/ d ν x (4.22) ? ? (where τ ν is the volume of the unit ball in R ν ). This was discovered in-dependently about the same time by Birman-Borzov [70], Martin [460],Robinson [538] and Tamura [709]. (Lieb and I only knew of Martin’swork when we looked at Thomas-Fermi although all but Tamura ex-isted at the time.) In Section 8, I’ll discuss what happens if V is not C ∞ ( R ν ).Using these ideas of dividing space into small boxes, it wasn’t hardto show that E Q ( V Z ) /E T F ( V Z ) → Z → ∞ if V ∈ C ∞ ( R ). Thesemethods don’t deal with the boxes around the nuclei at R j . Basically,one needs to show that the system doesn’t collapse on those points, i.e.most of the electrons wind up the boxes containing those points. WhenI left IHES for Marseille at the end of 1972, Lieb and I were at this pointand were left with the problem we called between ourselves “pulling thepoison Coulomb tooth”. I spent a long weekend in Paris in March, 1973and we figured out how to pull the tooth. With current technology, onewould use Lieb-Thirring inequalities (discussed in Section 8 below; seealso [446, 447] for the original papers, [312] for the discrete case and[679, Section 6.7] for a textbook discussion) but they didn’t exist, sowe used an adhoc argument exploiting the angular momentum barrier.The proof of (4.21) isn’t hard. The ρ ’s are functional derivatives ofthe energy under adding an infinitesimal V Z to the Coulomb attrac-tion. Normally convergence of functions doesn’t imply convergence ofderivatives but it does for concave functions and one can show the en-ergies, as minima of a set of functions linear in λ , are concave under λ → λV . WELVE TALES 31 I have one other result on large Z ions. As noted above, it is known(see [742, 583]) that the Hamiltonian, H ( Z, N ) for a charge Z nucleusand N electrons has infinitely many bound states if N ≤ Z . Whathappens if N > Z ? It is a result of Ruskai [549] and Sigal [568, 569]that there is a finite number N ( Z ), so that H ( Z, N ( Z )) has discretespectrum (i.e. there is a negative ion with nuclear charge Z and totalcharge − ( N ( Z ) − Z )) and so that for all N > N ( Z ), we have that N ( Z ) has no bound states below the continuum. In [439], Lieb, Sigal,Thirring and I showed that N ( Z ) /Z → Z → ∞ . That this isespecially subtle is seen by the fact that if one replace fermionic electonsby bosons (with negative charge), then Benguria-Lieb [64] have shownthat the analogous lim inf is strictly bigger than 1. I note that thereare no twice negatively charged ions known in nature so it is possiblethat N ( Z ) is always either Z or Z + 1. In fact, one of the fifteen openproblems in my 2000 list (Simon [658] of which 11 remain open) is toprove that N ( Z ) − Z remains bouned as Z → ∞ .Since 1973, there has been a huge literature on large Z atoms andmolecules and on density functional theory. I will not attempt acomprehensive review, but I should mention the work on non-leadingasymptotics beyond (4.20) and (4.21). For the energy, Hughes [308] andSiedentop-Weikard [566, 567] obtained the O( Z ) term for atoms andIvrii-Sigal [324] for molecules. Later, Fefferman-Seco [183] obtainedthe O( Z / ) term. For the density, Iantchenko-Lieb-Siedentop [319]found the O( Z ) term. Since much of the work on higher order correc-tion to (4.2) was done by Lieb and collaborators, I refer the reader tothe relevant volume of Lieb’s Selecta [438] for references. The readercan also look at two somewhat dated review articles by Lieb [437] andHundertmark [309].Before leaving this subject, I should mention that Lieb and I [443]used methods similar to those we used to prove existence of solutionsto the TF equation to prove existence of solutions to the Hartree andHartree-Fock equations for neutral (and positive) atoms and molecules.Later works on Hartee-Fock include Lions [449] and Lewin [431].5. Infrared Bounds and Continuous Symmetry Breaking (cid:104) s5 (cid:105) A fundamental problem in statistical physics concerns the following.The Gibbs states of statisitcal mechanics are clearly analytic in all pa-rameters, yet nature is full of discontinuities, for example the directionof a magnet as a magnetic field is slowly varied through zero field.We now realize that the way to understand this is by looking at thethermodynamic limit, i.e. infinite volume, where states can become non-analytic in parameters. That this is far from evident can be seenby a story told in Pais’ book [494, pp 432-433] that as late as 1937,at the van der Waals Centenary conference, a vote of the physicistspresent was taken on whether this view was correct and the vote wasclose (although, given that Peierls work mentioned below was in 1936,it shouldn’t have been close; that means that Peierls work was not wellknown, or at least not understood, at the time).The simplest models on which this can be explored are the latticegases whose formalism is described in the books of Ruelle [548] andSimon [646]. Two of the simplest examples both have spins on a lattice,say Z ν , σ α , at points α ∈ Z ν . In a finite box Λ ⊂ Z ν , the Hamiltonian(energy functional) is H Λ = − J (cid:88) α,γ ∈ Λ , | α − γ | =1 σ α · σ γ (5.1) The sign is there so that when J > 0, energies are lowest when spinsare parallel, i.e. the model is ferromagnetic. J < J = ± β in e − βH defining the Gibbs’measure. We have not been careful about boundary conditions; wewill most often take periodic BC although sometimes free or plus BC.We said two models because we haven’t described the set of singlespins and their distributions. If each σ α = ± Ising model . If instead each σ α ∈ S , the unit sphere is R with the rotation invariant aprioriweight, we have the classical Heisenberg model . More generally if each σ α ∈ S d − , the unit sphere is R d , we have the d -rotor model . Significanthere is the global symmetry of the system: discrete, σ α → − σ α for theIsing model and continuous, σ α → R σ α with R ∈ SO (3), the rotationgroup for the Heisenberg model.In 1936, Peierls [505] found a simple argument proving that when ν ≥ 2, the nearest neighbor Ising model on Z ν has a phase transition atlow temperature. However, the argument depends on the sharp differ-ence between spin up and spin down and fails for classicial Heisenbergmodel where the spins vary continuously. Indeed, in 1966, Mermin-Wagner [468] proved that in 2 D , the classicial Heisenberg model hasno broken symmetry states at positive temperature (see also [389]).Their argument relies on the fact that if the spin wave energy (when J = 1) is given by E p = 12 (cid:88) | α | =1 (1 − e ip · α ) = ν (cid:88) j =1 (1 − cos( p j )) (5.2) ? ? WELVE TALES 33 (the Fourier transform of the nearest neighbor coupling with a constantadded so that E p ≥ E p ∼ p for p smallso that (cid:90) | p j | <π d ν pE p = ∞ if ν = 2 (5.3) ? ? In 1976, Fr¨ohlich, Spencer and I (henceforth FSS) [203, 204] provedthat (cid:104) T5.1 (cid:105) Theorem 5.1 ([203, 204]) . The classical d -vector model ( d ≥ withnearest neighbor interactions on Z ν with ν ≥ has multiple phases(with broken symmetry) if β ≥ β c where β c ≤ β F SSc ≡ d I ( ν ) (5.4) ? ? with I ( ν ) = 1(2 π ) ν (cid:90) | p j | <π d ν pE p (5.5) so I ( ν ) < ∞ when ν ≥ I (3) can be computed exactly in terms of elliptic integrals so onefinds (with “errors” computed by comparing with high temperatureexpansions in parentheses) T c ( ν = 3 , d = 3) ≥ . . 44; 9% error) (5.6) ? ? and T c ( ν = 3 , d = 1) ≥ . . ? ? The method of FSS which I sketch below is basically the only methodknown for rigorously proving spontaneous continuous symmetry break-ing with a nonabelian symmetry group. We note that such continuoussymmetry breaking is not only central to statistical mechanics but alsoto models of particle physics.A basic notion is reflection positivity. This is one of the OsterwalderSchrader axioms mentioned in Section 3. FSS realized it also playedan important role in statistical mechanical models.Consider spins in a box, Λ with even sides with periodic boundaryconditions and slice the box across bonds into two halves, Λ + and Λ − .There is a natural reflection Θ of spins in Λ + onto spins in Λ − thatextends to a map of polynomials in the spins. A state (cid:104)·(cid:105) is called reflection positive (RP) if and only if for any polynomial, A , in thespins of Λ + we have that (cid:104) Θ( A ) A (cid:105) ≥ ? ? First, suppose that we consider uncoupled spins (i.e. a product mea-sure over sites). Then (cid:104) Θ( A ) A (cid:105) = |(cid:104) A (cid:105)| ≥ − H = A + Θ( A ) + (cid:88) j Θ( B j ) B j (5.9) ? ? We claim that if (cid:104)·(cid:105) is RP, so is (cid:104)·(cid:105) = (cid:104)· e − H (cid:105) / (cid:104) e − H (cid:105) for exp( A +Θ( A )) = exp( A )Θ(exp( A )) and we can expand exp( (cid:80) j Θ( B j ) B j ) intoa Taylor series.Consider a box, Λ, with periodic boundary conditions and state (cid:104)·(cid:105) Λ .Define, the magnetization (here and below we notationally suppress aΛ dependence). m = 1 | Λ | (cid:88) α ∈ Λ σ α (5.10) ? ? Our sign that there is a phase transition will be thatlim inf (cid:104) m (cid:105) Λ ≡ M > ? ? This implies many other notions of phase transition. For example,one can show that the derivative of the free energy per unit volumewith respect to an external magnetic field is has a discontinuity of atleast 2 M . Also, there are multiple equilibrium states in the sense ofDobrushin [154, 155] and Lanford-Ruelle [411] (see [646, Section III.2]).We let Λ ∗ be the dual lattice to Λ so that { α (cid:55)→ | Λ | − / e i p · α } p ∈ Λ ∗ is an orthonormal basis for the (cid:96) (Λ). Define the Fourier spin wavevariables (cid:98) σ p = 1 (cid:112) | Λ | (cid:88) α ∈ Λ e − ip · α σ α (5.12) ? ? and define the spin wave expectation function g Λ ( p ) = (cid:104) (cid:98) σ p · (cid:98) σ − p (cid:105) (5.13) ? ? = 1 | Λ | (cid:88) α,β ∈ Λ e − i p · ( α − β ) (cid:104) σ α · σ β (cid:105) = (cid:88) α e − i p · α (cid:104) σ α · σ (cid:105) (5.14) ? ? Note that m = | Λ | − / (cid:98) σ p =0 so that (cid:104) m (cid:105) Λ = | Λ | − g Λ ( p = 0) (5.15) WELVE TALES 35 Since (cid:98) σ p are components of the functions α (cid:55)→ σ α in an ON basis, thePlancherel theorem implies that (cid:88) p ∈ Λ ∗ g Λ ( p ) = (cid:88) α (cid:104)| σ α | (cid:105) = | Λ | (5.16) ? ? Since there are | Λ | values of p in Λ ∗ , this says that normally each g Λ ( p )should be of size 1 while the condition of there being a phase transitionis that g Λ ( p = 0) is of order Λ. This allows one to interpret the phasetransition as due to a Bose condensation of spin waves.The key to the proof will be what FSS dubbed an infrared bound (IRB), that for p (cid:54) = 0, one has that: g Λ ( p ) ≤ d βE p (5.17) where E p is the spin wave energyBy (5.15)-(5.17) lim inf | Λ |→∞ (cid:104) m (cid:105) Λ ≥ − dI ( ν )2 β (5.18) ? ? where I ( ν ) is given by (5.5) and we use the fact that Λ ∗ fills out the ν -fold product of [ − π, π ] as | Λ | → ∞ . Thus infrared bounds implyTheorem 5.1.The first step in the proof of infrared bounds from reflection pos-itivity is to use RP to prove something called Gaussian domination,namely if we define, for arbitrary { h α } ∈ R d | Λ | Z ( { h α } ) = (cid:90) S ( d − | Λ | exp − β (cid:88) | α − γ | =1 ( σ α − σ γ − h α − h β ) (5.19) ? ? then one has that Z ( { h α } ) ≤ Z ( { h α } ≡ 0) (5.20) One first proves that if Λ is split into two halves Λ + and Λ − and,given h , we let h + be the H obtaining by restricting h to Λ + andreflecting it and similarly for h − , then Z ( { h } ) ≤ Z ( { h + } ) / Z ( { h − } ) / (5.21) ? ? The details of the proof of (5.20) can be found in [204] or [200].Once one has Gaussian domination, fix h all real and use the fact that Z ( { λ h } ) is maximized at λ = 0 so the second derivative is negative. This implies that (cid:42) (cid:88) | α − γ | =1 | ( σ α − σ γ ) · ( h α − h γ ) | (cid:43) Λ ≤ β (cid:88) | α − γ | =1 | h α − h γ | (5.22) ? ? Adding the results for the real and imaginary part extends this inequal-ity to complex h . Taking h to be a plane wave with a single component(and summing over possible components) proves the Infrared boundand completes the proof that RP ⇒ Phase Transition.A little about the history of this work with Fro¨ohlich and Spencer.In October, 1975 I heard indirectly that J¨urg and Tom had found thatthere was spontaneously symmetry breaking in the multi-componentΦ EQFT. The analog of infrared bounds for this case was easy. Thereis a K¨allen-Lehman representation for the two point function (cid:104) ϕ ( x ) · ϕ ( x ) (cid:105) = α + (cid:90) dρ ( m ) (cid:90) d k (2 π ) e ik · ( x − y ) ( k + m ) − (5.23) ? ? where if ϕ has N components, one has that (cid:90) dρ ( m ) = N (5.24) The infrared bound then just needs that ( k + m ) − ≤ k − . Since I’dheard of this indirectly and they were looking at the field theory, I felt,perhaps unfairly, that I could think about the statistical mechanicalanalog. I realized that the key was (5.24) which followed from canonicalcommutation relations and I found a commutation inequality for thetransfer matrix and could use that to push through a phase transition(this is the argument that appears in [203]). The three of us met at theAMS meeting in San Antonio in Jan 1976 and agreed to publish ourresults jointly. We found the Gaussian domination approach duringthe writeup of the full paper.It is not surprising that the work on infrared bounds generated con-siderable further work (479 Google scholar citations). I will not try todescribe all of it but will focus on two further developments in whichI played a role. The first concerns phase transitions in spin systemswith long range interactions. It was known for many years that finiterange 1 D systems could not have phase transitions (see, for example[646, Theorem II.5.3]). Ruelle [547] proved that this remains true forinfinite range interactions with not too slow decay. In particular, forthe pair interacting ferromagnetic model with J ( n ) = (1 + | n | ) − α , heproved there were no phase transitions if α > WELVE TALES 37 On the other hand, Dyson [165], exploiting correlation inequaltities,proved there are phase transitions if 1 < α < 2. For 2 D , as we’veseen Mermin-Wagner [468] proved that plane rotors had no symmetrybreaking for nearest neighbor interactions but Kunz-Pfister [401] usedDyson’s method to prove that 2D plane rotors with a similar pair inter-action has a broken symmetry phase transition if 2 < α < 4. Becausethese results depend on correlation inequalities which fail for classicalHeisenberg models, the proofs do not extend to such Heisenberg mod-els. Indeed, Dyson conjectured but could not prove that his result wasstill true in the Heisenberg case.With Fr¨ohlich, Israel and Lieb, I showed [199] infrared bounds couldbe proven for such long range models and, in particular, we provedDyson’s conjecture. We called a function J on Z + , the strictly positiveintegers, RP if and only if for all positive integers, n and z ∈ C n , onehas that (cid:88) i,j ≥ ¯ z i z j J ( i + j − ≥ Then FILS show that 1D ferromagnets with − H = (cid:80) i Theorem 5.2 ([167]) . The nearest neighbor quantum Heisenberg anti-ferromagnet for s ≥ and ν ≥ and for spin / and sufficiently large ν has a phase transition with N´eel order. Remarks. 1. N´eel order means that for any Λ with even sides, onehas that lim inf (cid:28)(cid:16) | Λ | (cid:80) α ∈ Λ ( − | α | σ α (cid:17) (cid:29) Λ > / ν = 3.3. While the bounds on β c are concrete, they involve an implicitequations which includes the ground state energy of the antiferromag-net which is not known in closed form.There are two issues involving the quantum case viz a viz the clas-sicial case that should be mentioned. First infrared bounds cannothold in the form of (5.17) because they imply that as β → ∞ that (cid:104) σ α · σ γ (cid:105) goes to a constant (i.e. independent of α and γ ). Inthe classical case, this quantity goes 1 uniformly in the sites (for Λfixed). But in the quantum case with spin S , it goes to S ( S + 1)for α = γ but only to S for α = γ (for the maximum spin value of σ α + σ γ is 2 S so the maximum value of σ α · σ γ = [( σ α + σ γ ) − σ α − σ γ ] = [2 S (2 S + 1) − S ( S + 1)] = S ). The solution isto get an initial inequality not on the thermal expectation (cid:104) AB (cid:105) =Tr( ABe − βH ) / Tr( e − βH ) but what DLS call the DuHamel two point func-tion ( A, B ) = (cid:82) Tr( e − xβH Ae − (1 − x ) βH B ) / Tr( e − βH ). Since DLS provethat ( A ∗ , A ) ≥ g ( A ) f ( c ( A ) / g ( A ) where g ( A ) = (cid:104) A ∗ A + AA ∗ (cid:105) , c ( A ) = (cid:104) [ A ∗ , [ H, A ]] (cid:105) and f is the function given implicitly by f ( x tanh x ) = x − tanh x and this implies a direct bound with coth, various formulainvolving coth occur, indeed, they conjecture (but do not prove) thatthe correct analog of (5.17) is g Λ ( p ) ≤ (cid:113) S coth (cid:16)(cid:113) SβE p (cid:17) where g Λ ( p ) is thermal expectation of (cid:98) σ p · (cid:98) σ − p .Secondly, to get infrared bounds on the DuHamel functions, oneneeds that the algebra of matrices on which the reflection acts in non-commutative RP to be real matrices. Of course, the usual representa-tion of Pauli spins is not real. σ and σ are but σ = (cid:0) i − i (cid:1) is not!Indeed, because of the commutation relations [ σ , σ ] = 2 iσ , there isno representation in which all spins can be real. For the antiferromag-net, one can take s = σ , s = iσ , s = σ and let the reflection WELVE TALES 39 be Θ(( s α ) j ) = (cid:26) − ( s Rα ) j , j = 1 , s Rα ) j , j = 2 (5.26) ? ? so that − σ α · σ Rα = s α · Θ( s α ) and so get positivity under a reflectionon a real algebra for the antiferromagnet. The corresponding infraredbound on the DuHamel two point function then reads (cid:0) ( (cid:98) σ p ) j , ( (cid:98) σ − p ) j (cid:1) ≤ (cid:101) E p (cid:101) E p = ν + ν (cid:88) cos( p j ) (5.27) ? ? (the sign in (cid:101) E p is such that it vanishes at p j = π consistent with N´eelorder). This bound leads to Theorem 5.2.Dyson, Lieb and I initially thought that we had a trick for getting realmatrices for the ferromagnet. One can double dimension and replacemultiplication by i by (cid:0) − (cid:1) and thereby homomorphically map n × n complex matrices to a subset of all 2 n × n real matrices. However itturns out when you do this at each site for Pauli matrices, the naturalreflection no longer has reflection positivity. Our announcement [166]focused on the ferromagnet and so did the preprint of [167]. However,Fr¨ohlich was giving a course at Princeton on the work of FSS and DLSand didn’t understand one step in our preprint. He found this and cameto us on the same day we’d finishing correcting the galley proofs for thelonger article; indeed, after we placed the envelope with them in theoutgoing departmental mailbox. We immediately realized that therewas a problem and retrieved and then fixed the galley proofs so that thepublished version of [167] is correct. I note that now, almost 45 yearslater, there is no rigorous proof of the existence of a phase transitionin the quantum Heisenberg ferromagnet. It is fortunate that none ofus was a young unknown when this work was done for while there is acorrect very important result, the wrong result was embarrassing. Thepaper [167] does have results on the quantum xy ferromagnet where thecoupling drops the σ α,z σ β,z (the xy model has an abelian continuoussymmetry); this is possible because there is a representation in whichtwo of the Pauli matrices are both real.There has been considerable literature on the quantum Heisenbergmodel since. There is a lovely online bibliography on this subject postedby Kennedy-Nachtergale [373].Another application of RP methods involves what is called the Chess-board Peierls Method . Fr¨ohlich, Israel, Lieb and I wrote two papers[199, 200] that systematized both infrared estimates and this methodand, in particular, applied the Chessboard Peierls method to a number of models. The key is what is called Chessboard Estimates . The namewas introduced by Fr¨ohlich-Simon [202] in a paper in the Annal ofMathematics on the structure of states in general P ( ϕ ) quantum fieldtheories. They could not use the less fancy term “checkerboard esti-mate” because that had already been used by GRS [274] for a differentbound.While FS systematized the estimates and introduced the name,the idea had appeared earlier in works of Glimm-Jaffe Spencer [241],Guerra[270], Seiler-Simon [563] and Park [496, 497]. Fr¨ohlich-Lieb(henceforth FL) [201] following up on their use in QFT by Glimm-Jaffe-Spencer and exploited these estimates with the Peierls argumentto prove phase transitions in spin models and this was pushed furtherby FILS.We consider a box Λ, typically with periodic BC, that can be parti-tioned by hyperplanes into boxes, { ∆ α } α ∈ Q , so that there are an evennumber of boxes in each direction and so that there is RP in each hy-perplane. Given a function F of the spins in box ∆ α , we cover Λ bycontinually reflecting F in hyperplanes and let γ ( F, ∆ α ), be the | Λ | th root of the (cid:104)·(cid:105) Λ expectation of the product of these reflected copies of F .The chessboard estimate says that given functions, F α , of the spinsin ∆ α , one has that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:42) (cid:89) α ∈ Q F α (cid:43) Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:89) α ∈ Q γ ( F α , ∆ α ) (5.28) ? ? If the number of edges in each direction is a power of 2, it is easy toprove the estimate directly by multiple use of the Schwarz inequality.In general, one uses an argument reminiscent of the proof of Gaussiandomination (in fact, one can prove Gaussian domination from Chess-board Estimates). One considers the ratio of the two sides of the Chess-board estimate as each F α runs through the various F ’s and their reflec-tions, considers the one that maximizes it and then uses RP to proveamong the maximizers is one where the ratio is 1.The Peierls strategy sums on contours that separate various statesof the system. A key part of the strategy is the estimation of theprobability of large contours. Those can typically be thought of ex-pectations of products of bad events, typically one for each link in thecontour. One can use a checkerboard estimate to get upper boundson these probabilities in terms of thermodynamic quantities and thisis the Chessboard Peierls method. WELVE TALES 41 In particular, FL showed this approach was effective in studyinganisotropic classical Heisenberg models; they succeeded in proving aphase transition in 2 D for arbitrarily small anisotropy. FILS used thistechnique in a wide variety of models including ones with no symmetry.In particular, FILS recovered results of the Piragov Sinai [509, 510]approach in different way.Since this section is the only one on statistical mechanics, per se, Iend it with a brief discussion of some of my other work in the subject.First a paper [619] (and a brief report in [617]) on the classical limitof quantum spin models. This was motivated by a wonderful paperof Lieb [433] who considered a classical Hamiltonian, H Λ ( { σ α } α ∈ Λ ),which is affine in the spins σ α ∈ S , α ∈ Λ. The classical partitionfunction is Z cl ( γ ) = (cid:90) (cid:89) α ∈ Λ [ d Ω( σ α ) / π ] exp( − H Λ [ { σ α } ]) (5.29) ? ? where d Ω is the usual unnormalized measure on the unit sphere, S , in R . For (cid:96) = , . , . . . , define Z (cid:96)Q ( γ ) = (2 (cid:96) + 1) −| Λ | Tr(exp[ − H ( γL α /(cid:96) )]) (5.30) ? ? where L α is an independent spin (cid:96) quantum spin at each site α ∈ Λ.Then Lieb proved that Z cl ( γ ) ≤ Z (cid:96)Q ( γ ) ≤ Z cl ( γ + (cid:96) − γ ) (5.31) Among other things, this immediately implies convergence of Z (cid:96)Q ( γ ) to Z cl ( γ ) as (cid:96) → ∞ because Z cl ( γ ) is continuous in γ (indeed it is ana-lytic). Moreover, in situations where one knows that the infinite volumelimit object (the pressure), p · ( γ ) = lim | Λ | Z · , Λ ( γ ) exists, it implies con-vergence of the limit objects, since while they might lose analyticity inthe limit, they are convex and so continuous.I had begun teaching a course on group representations which even-tually turned into a book [653] and it occurred to me to wonder whatthe analog of (5.31) was if the representations of SU (2) or SO (3) werereplaced by a more general compact Lie group. In particular, whatclassical limt space replaces S .While it was mathematical elegance that attracted me, I had an-other motivation. Dunlop-Newman [161] had proven a Lee-Yang zerotheorem for S spins by using the fact that Asano [19] had proven onefor spin quantum spins, the Griffiths trick [264] then gets it for spin (cid:96) quantum spins and the limit theorem implied by (5.31) then implies one for S spins. It was natural to worry about spins on S d − (i.e. d -component rotors).Since I eventually showed the classical limit spaces are symplecticmanifolds, S N is never a classical limit if N ≥ S d − for all d to proving a conjecured Asano type result for spin SO (2 k ) spinors.To this day, not only is that conjecture still open but so is whetherLee-Yang holds for S d − spins with d ≥ λ , on a compact Lie group, G ,and for each L = 1 , , . . . , one considers the irreducible representation, U L , on H L , with maximal weight Lλ . By picking a basis in the Liealgebra, g , of G , one considers Hamiltonians multilinear in the basisvectors at the various sites in a box Λ. If d L = dim( H L ), then Z LQ ( γ ) = d −| Λ | L Tr(exp[ − H ( γS α /L )]) (5.32) ? ? Extend λ to g ∗ , the dual of g by setting it to zero on the orthogonalcomplement of the Cartan subalgebra. Under the dual of the adjointaction of G on g , one gets a manifold by looking at the orbit of thisextention of λ . These coadjoint orbits, Γ λ , which also plays a role inthe Kirilov [375] theory of representations of nilpotent Lie groups andin the closely related Kostant [392]-Sourieau [696] method of geomet-ric quantization, are the classical limts. Haar measure on G inducesa measure on Γ λ and one uses this to define a suitable classical limitpartition function Z cl ( γ ). By using coherent vectors based on the max-imal weight vectors and the same Berezin-Lieb inequalities that Liebdid, I could extend his result to this case.There is a magic weight, δ , which is the sum of all the fundamentalweights. Let a = 2 (cid:104) λ, δ (cid:105) / (cid:104) λ, λ (cid:105) where (cid:104)· , ·(cid:105) is the Killing inner producton the weight space. Then I extend (5.31)to Z cl ( γ ) ≤ Z LQ ( γ ) ≤ Z cl ( γ + aL − γ ) (5.33) SU (2) is rank 1, so there is a single fundamental weight and δ = λ so a = 2. Moreover, L = 1 corresponds to (cid:96) = , so (cid:96) = L . Thus,(5.33) in this case is just (5.31). One surprise of this analysis is thatthere are several distinct classical limit spaces if the rank is 2 or more.For example, for SO (4), the space for the limit of spherical harmonicsis the 4-d space S × S while for the spinor representations, it is the2-d space S ∪ S . WELVE TALES 43 At Princeton, I ran a “brown bag seminar” which included brief pre-senations about current research both on one’s own work and work ofothers. There were typically about 25 participants that often includedall the senior math physics faculy at Princeton (Lieb, Nelson, Wight-man and me) and Dyson from the Institute as well as our wonderfulgroup of postdocs/junior faculty/grad students (see [675] for a completelist but included were Aizenman, Avron, Fr¨ohlich, Deift and Sigal). Inthe fall of 1979, Michael Aizenman came back from a conference inHungary and, at a brown bag, reported on some work of Dobrushin-Pecherski (a small part of [156]) that showed sufficiently fast powerdecay of correlations in spin systems implied exponential decay. Intrying to understand why this might be, I proved the following: ? (cid:104) T5.3 (cid:105) ? Theorem 5.3 ([621, 620]) . Let (cid:104) σ α σ γ (cid:105) denote the two point functionof a spin nearest neighbor (infinite volume, free boundary condition)Ising ferromagnet at at some fixed temperature. Fix α, γ and B , a setof spins whose removal breaks the lattice in such a way that α and γ lie in distinct components. Then: (cid:104) σ α σ γ (cid:105) ≤ (cid:88) δ ∈ B (cid:104) σ α σ δ (cid:105)(cid:104) σ δ σ γ (cid:105) (5.34) Remarks. 1. One consequence of this is that if the lattice is Z ν , thenif (cid:104) σ α σ γ (cid:105) ≤ C | α − γ | − µ with µ > ν − 1, then for some C and m > (cid:104) σ α σ γ (cid:105) ≤ C e − m | α − γ | .2. I talked about this at a later brown bag which stimulated addi-tional work: Lieb found an improvement and Aizenman and I founda version for multicomponent models. We arranged for these threepapers to appear successively in CMP. Lieb’s improved result [436] in-volved the component Λ of Z ν \ B with α ∈ Λ and allowed (cid:104) σ α σ δ (cid:105) in (5.34) to be replaced by (cid:104) σ α σ δ (cid:105) B ∪ Λ , the expectation with interac-tions outside B ∪ Λ dropped. This included Griffiths third inequality[263]. These geometric correlation inequalities are sometimes calledLieb-Simon inequalities as a result.3. As mentioned, Aizenman and I [10] proved a version for d -vectormodels.4. Related inequalities appeared earlier in work of Kastelyn-Boel[349].Among some of my other results on lattice gases are1. A work with Sokal [685] which made rigorous an argument ofThouless [715] exploiting energy-entropy estimates, that, for example,provided another proof of the result of Ruelle [547] that a pair of spin Ising ferromagnets whose coupling obeys (cid:80) n | J ( n ) | < ∞ has zerospontaneous magnetization.2. A paper [625] on the one dimensional d -rotor model with critical J ( n ) = | n | − (for n ≥ J ( n ) = | n | − α Dysonshowed phase transitions for Ising spins if 1 < α < 2, FILS provedphase transitions for d -rotor models with 1 < α < α > 2. The case α = 2 is borderline. Fr¨ohlich-Spencer [205] proved Ising models with this borderline α have discretesymmetry breaking. In [625], I proved d -rotor models with d ≥ d -rotor models, mean field theoryprovides upper bounds on transition temperatures.6. A book [646] that discusses the lattice models that have been atthe center of this section. It focuses on formalism and does not discusscorrelation inequalities, Lee-Yang, the Peierls argument and infraredbounds, some of the most fascinating aspects of the subject. I have abook on those aspects of this subject in preparation [683].6. N –Body quantum mechanics (cid:104) s6 (cid:105) The previous sections have focused on one or two problems, all (butthe last section) within a limited area of mathematical physics. This isa much more diffuse section dealing with general N -body NRQM so Iwill leave more background to references and only briefly discuss a lotof work. In particular, by thinking of 2 as a possible value of N , I’llthrow in some subjects that are not usually considered N -body QMlike some inverse potential scattering and even a little bit of general1 D Schr¨odinger operators.A full N -body Hamiltonian acts on L ( R νN ) where x ∈ R νN is writ-ten x = ( r , . . . , r N ) with r j ∈ R ν . We write (cid:101) H = − N (cid:88) j =1 (2 m j ) − ∆ r j (cid:101) V = (cid:88) ≤ i T6.1 (cid:105) WELVE TALES 45 Theorem 6.1. In any coordinate system, ρ , . . . , ρ N where ρ j , j =1 , . . . , N − is a linear combination of r k − r (cid:96) and ρ N = 1 M N (cid:88) j =1 m j r j (6.2) ? ? we have that, realizing (cid:101) H ≡ L ( R νN ) = L ( R ν ) ⊗ L ( R ν ( N − ) ≡ H CM ⊗H , where the first factor is functions of ρ N and the second functions of { ρ j } N − j =1 , (cid:101) H = h ⊗ + ⊗ H (6.3) (cid:101) H = h ⊗ + ⊗ H (6.4) ? ? where h = − (2 M ) − ∆ ρ N , H is a positive quadratic form in − i ∇ ρ j , j = 1 , . . . , N − and H = H + V . I refer the reader to [682, Section 11] for a discussion of variouscoordinate systems and the formalism of Sigalov–Sigal [578] (see alsoHunziker–Sigal [318]). In that formalism, a major role is played by theinner product (cid:104) r (1) , r (2) (cid:105) = N (cid:88) j =1 m j r (1) j · r (2) j (6.5) In this inner product, (cid:101) H is the Laplace-Beltrami operator and thereason that (6.3) holds is that ρ N is orthogonal to the other ρ j ’s. Onecoordinate system that we’ll need soon is atomic coordinates where ρ j = r j − r N , j = 1 , . . . , N − ρ N = 1 M N (cid:88) j =1 m j r j (6.6) In this coordinate system when m = m = · · · = m N − = m and m + m N ≡ µ , one has that (see [682, (11.48)] for the calculation) H = − N − (cid:88) j =1 µ ∆ j − m N (cid:88) j 2. We set C (cid:96) ) to be thenumber of particles in C (cid:96) . A coordinate, ρ , is said to be internal to C (cid:96) if it is a function only of { r m } m ∈ C (cid:96) and is invariant under r m → r m + a ,equivalently, it is a linear combination of { r m − r q } m,q ∈ C (cid:96) . A clusteredJacobi coordinate system is a set of C (cid:96) ) − R (cid:96) = ( (cid:80) q ∈ C (cid:96) m q r q ) / ( (cid:80) q ∈ C (cid:96) m q ),If we write H ( C (cid:96) ) to be L of the internal coordinates of cluster C (cid:96) and H ( C ) to be L of all the centers of mass of the clusters then (cid:101) H = (cid:101) H ( C ) ⊗ k (cid:79) (cid:96) =1 H ( C (cid:96) ) (6.8) (cid:101) H = (cid:101) H ( C )0 ⊗ · · · ⊗ + k (cid:88) (cid:96) =1 ⊗ · · · ⊗ H ( C (cid:96) ) ⊗ · · · ⊗ (6.9) where (cid:101) H ( C )0 = − (cid:80) k(cid:96) =1 (2 M ( C (cid:96) )) − ∆ R (cid:96) and H ( C (cid:96) ) is a quadratic formin the derivatives of the internal coordinates.In (6.9), the operator (cid:101) H ( C )0 has a decomposition like (6.3) where H isreplaced by H ( C ) , the functions of the differences of the centers of massof the C j . We write (cid:101) H ( C )0 = h ⊗ + ⊗ H ( C )0 (6.10) ? ? Given a cluster decomposition, C = { C (cid:96) } k(cid:96) =1 , we write ( jq ) ⊂ C if j and q are in the same cluster of C and ( jq ) (cid:54)⊂ C if they are in differentclusters. We define V ( C (cid:96) ) = (cid:88) j,q ∈ C (cid:96) j C ∈ P ⇒ σ ( H ( C )0 ) = [0 , ∞ ) (6.17) By (6.15), we have that (where σ ( · ) is the spectrum) σ ( H ( C )) = σ ( H ( C )0 ) + σ ( H ( C )) + · · · + σ ( H ( C k )). By (6.17) C ∈ P ⇒ σ ( H ( C )) = [Σ( C ) , ∞ ) (6.18) ? ? When we discuss N –body spectral and scattering theory below, we’llbe interested in thresholds. A threshold , t , is a decomposition C = { C (cid:96) } k(cid:96) =1 ∈ P and an eigenvalue, E (cid:96) of H ( C (cid:96) ) for each (cid:96) = 1 . . . . , k . The threshold energy is E ( t ) = (cid:80) k(cid:96) =1 E (cid:96) . Of course, E ( t ) ≥ Σ( C ).With these preliminaries in hand, we can describe the central math-ematical questions in the analysis of N -body NRQM. We assume thereader is familiar with the basic notions of self-adjointness of un-bounded operators, the spectral theorem for them [680, Chapters 5 and7] and the spectral decompositions into discrete and essential spectrumand into absolutely continuous, singular continuous and point spectrum(see [680, Theorem 5.1.12]). One always supposes that the two bodypotentials, V ij , go to zero at infinity, usually faster than r − − ε .(1) The self-adjointness of H .(2) The determination of the essential spectrum of H . Thisis solved by the celebrated (cid:104) T6.2 (cid:105) Theorem 6.2 (HVZ Theorem) . For reduced N body Hamiltonians withtwo body potentials vanishing at infinity, one has that σ ess ( H ) = [Σ , ∞ ) Σ = inf C∈ P Σ( C ) (6.19) ? ? (3) Absence of singular continuous spectrum for H . My ad-visor, Arthur Wightman had a colorful name for this: “the no goohypothesis”. His point was that a.c. spectrum had the interpretationof scattering states and point spectrum as bound states. If there weresingular continuous spectrum, it would have to be goo. Connected tothis is that point spectrum should only have limit points at thresholds;one might expect no embedded point spectrum but the examples dis-cussed at the start of Section 2 show that is too simple minded althoughone might like to prove the absence of positive energy eigenvalues andthresholds. (4) Asymptotic Completeness . To describe this, we need someadditional preliminaries. Let t be a threshold and C the associatedcluster decomposition. Under the decomposition (6.8), we let H t beall states of the form ϕ ⊗ η where ϕ is an arbitrary vector in (cid:101) H ( C ) (i.e.function of the differences of centers of mass of the clusters) and η a sumof products of eigenvectors of H ( C (cid:96) ) with eigenvalue E (cid:96) . Let P t be theprojection onto H t . In 1959, Hack [277] proved that, for each thresholdthe limits (the funny convention that has Ω ± associated to limits as t → ∓∞ comes from the physics literature where Ω ± defined this wayis connected in time independent scattering to lim ± ε ↓ ( H − E − iε ) − ).Ω ± t = s − lim t →∓∞ e itH e − itH ( C ) P t (6.20) ? ? exists so long as the two body potentials decay faster that r − − ε (forlonger range, including the physically important Coulomb case, fol-lowing Dollard [157], one needs to use modified wave operators - see,for example [527, Section XI.9] - we’ll refer to this case below with-out further technical detail). These are the cluster wave operators .If ψ = Ω − t γ , then as t → ∞ , we have that e − itH ψ looks like boundclusters of C (cid:96) in eigenstates with energy E (cid:96) moving freely relative toeach other, i.e. intuitively scattering states. One can show that fordistinct thresholds, t (cid:54) = s , one has that ran Ω − t is orthogonal to ran Ω − s .Asymptotic completeness is the assertion that (cid:77) all thresholds t ran Ω + t = (cid:77) all thresholds t ran Ω − t = H ac ( H ) (6.21) ? ? where H ac ( H ) is the space of all vectors whose spectral measures for H is purely absolutely continuous.For each of these four, I made significant, albeit not the definitive,contributions as I’ll describe soon. Kato (see [682, Sections 7-10]) wasboth the pioneer and continuing master of the self-adjointness problembut I made a basic discovery on allowed local positive singularities andfollowed up on Kato’s work on what I called Kato’s Inequality. I notonly named the HVZ theorem (where Hunziker, van Winter and Zhislinwere the initiators) but reworked and extended it twice, including mywork with Last on the ultimate HVZ theorem. Perry, Sigal and I werethe first to prove the absence of singular continuous spectrum for fairlygeneral N -body operators although we relied heavily on ideas of Mourre(and Balslev-Combes had earlier handled suitable analytic potentialsincluding the important Coulomb case). Sigal-Sofer were the first toestablish N -body asymptotic completeness but they (and later, others)relied in part on my work with Deift which reduced the problem to theexistence of what are now called Deift-Simon wave operators. WELVE TALES 49 In the remainder of this section, I’ll discuss in more detail these andother works on N -body and related problems. My earliest paper ongeneral N-body systems is Simon [583] which proved that general atomsand postive ions have infinitely many eigenvalues (aka bound states)below their continuous spectrum. This is really a remark on a paperof Kato [354] written 20 years earlier. In that paper, Kato showedthat Helium in the approximation of infinite nuclear mass had infin-itely many eigenvalues and with its physical mass has at least 25,585eigenvalues (counting multiplicity). Kato could not go beyond Heliumand had the 25,585 limitation because he only used crude methods toestimate the bottom of the continuous spectrum. The basic point of[583] is that since Kato’s work, Hunziker [316] had proven Theorem 6.2above and that by using that, it was not difficult to exploit the methodof Kato in [354] to get the very general result. I should mention thatten years earlier, Zhislin [742] using more involved methods had proventhis general result, so my proof was new, but the result was not.I knew about Hunziker’s paper [316] because he had done the workwhile a postdoc at Princeton and it was something that my advisorArthur Wightman discussed in his course. To understand the basis ofHunziker’s proof, it pays to recall the essence of one argument for thereduced 2-body case: if H = − ∆ , H = H + V and V goes to zero atinfinity, then σ ess ( H ) = [0 , ∞ ). One writes down the second resolventequation:( H − z ) − = ( H − z ) − − ( H − z ) − V ( H − z ) − ⇒ ( H − z ) − = ( H − z ) − (cid:2) V ( H − z ) − (cid:3) − (6.22) ? ? Since V goes to zero, z (cid:55)→ V ( H − z ) − is a compact analytic functionon C \ [0 , ∞ ), so, by the analytic Fredholm Theorem ([525, TheoremVI.4], [680, Theorem 3.14.3]), [1 + V ( H − z ) − ] − is meromorphic on C \ [0 , ∞ ) with finite rank residues. This implies the claimed result on σ ess ( H ). Hunziker instead used the fact that Weinberg [730] and vanWinter [724], essentially by resuming perturbation theory, proved theWeinberg-van Winter equations( H − z ) − = D ( z ) + ( H − z ) − I ( z ) (6.23) ? ? where D ( z ) and I ( z ) are built out of the potentials and the resolventsof the H ( C ) and so analytic in C \ [Σ , ∞ ). Moreover, Hunziker [315]proved that I ( z ) was compact (this was proven by Weinberg when N = 3 and conjectured in general) so, as in the reduced 2-body case,one gets the full N body result. Over the next few years, I became aware that in [724], van Winter(who like Hunziker specifically looked at ν = 3 but further restricted to V ij ∈ L ( R ) so she could use Hilbert Schmidt rather than just compactoperators) implicitly had Theorem 6.2 when her conditions hold by amethod close to Hunziker’s. Moreover, Zhislin [742] had the resultfor atoms using very different, geometric methods, but his method, asexplicated by J¨orgens-Weidmann [345], could also obtain Theorem 6.2.Thus, by the time of Reed-Simon, vol. 4 [528], I had decided to callthe result the HVZ theorem, a name which stuck.I provided two generations of improvements in this result and itsproof. In Simon [608], motivated in part by work on Deift-Simon waveoperators (see below), I found a geometric way of understanding thetheorem (independently, Enss [171], at about the same time, found aproof similar in spirit). Given a partition, C , one defines | r | C = min ( jq ) (cid:54)⊂C {| r j − r q |} | r | = max j (cid:54) = q {| r j − r q |} (6.24) One constructs a C ∞ partition of unity (i.e. (cid:80) C∈ P j C = 1 , j C ≥ 0) sothat for some d N > j C is supported on { r | | r | ≤ } ∪ { r | | r | C ≥ d N | r |} . [117] call this a Ruelle-Simon partition of unity since Ruelleconstructed such partitions in his work on QFT scattering theory. Ishowed easily that if f is a continuous function of compact support,[ f ( H ) − f ( H ( C ))] j C is compact. This is because I ( C ) j C decays in alldirections. One then writes f ( H ) = (cid:88) C∈ P [ f ( H ) − f ( H ( C ))] j C + (cid:88) C∈ P f ( H ( C )) j C (6.25) ? ? to conclude that if f is supported on ( −∞ , Σ), then f ( H ) is compactwhich implies Theorem 6.2.Many years later, in 2006, Last and I [416] returned to this subjectin a much more general context. I’ll state our result for Schr¨odingeroperators, − ∆ + V , on L ( R γ ) when V is uniformly continuous (whichis true for N -body systems if all V ij are continuous and go to zero atinfinity). We need the notion of limit at infinity. By the Arezel`a-AscoliTheorem [676, Theorem 2.3.14], V ( · + y ) restricted to large balls lies ina compact set as y varies through R γ . It follows that for any y m goingto infinity, there is a subsequence y m j so that V ( · + y m j ) converges tosome W uniformly on compact subsets of R γ . If y m j / | y m j | → e ∈ S γ − ,we say that W is a limit of V at infinity in direction e . We let L e be theset of such W . By the compactness noted above, each L e is non-empty.Here is what Last and I [416] proved. (cid:104) T6.3 (cid:105) WELVE TALES 51 Theorem 6.3. For any Schr¨odinger operators, − ∆ + V , on L ( R γ ) with V that is uniformly continuous, one has that σ ess ( − ∆ + V ) = (cid:91) e ∈ S γ − (cid:91) W ∈L e σ ( − ∆ + W ) (6.26) Remarks. 1. [416] also has results for Schr¨odinger operators where V is allowed to have local singularities (stated in terms of uniformlylocal Kato class) and for Jacobi matrices, CMV matrices and, as I’llmention in the next section, for Schr¨odinger operators with magneticfield.2. The proof is really quite simple based on Weyl sequences [680,Problem 3.14.5], i.e. if A is self-adjoint, then λ ∈ σ ess ( A ) if and only ifthere exists a sequence of unit vectors, { ϕ n } going weakly to zero with (cid:107) ( A − λ ) ϕ n (cid:107) → 0. We used localization ideas going back to Sigal [568]and G˚arding [210] to show one could pick the Weyl sequence to live ina large ball (of n independent size) and then compactness to get a trialsequence for a limit at infinity.3. Earlier in [415], Last and I had introduced the notion of right limitfor Schr¨odinger operators on the half line and proven that in that casethe right side of (6.26) is a subset of the left side. Limits at infinitygeneralize the notion of right limit. [415] also have results relatingright limits and a.c. spectrum which were generalized in a beautifuland spectacular way by Remling [534]. The work of Last-Simon [416]and Remling [534] is presented in [674, Chapter 7]. By exploitinganalogy, Breuer and I [78] used Remling’s idea as an organizing toolin understanding an issue in classical complex analysis: which powerseries lead to natural boundaries on their disk of convergence.4. Forms of (6.26) seem to have been in the air after 2000. Asdiscussed in Last-Simon [416] and [674, Chapter 7] (where referencescan be found) several other groups from very different communitiesfound variants of (6.26). Their proofs used much more machinery than[416]. In particular, [416] required the closure of the set of the rightside of (6.26) but using ideas of Georgescu-Iftimovici [213], one canshow that the set is closed.5. (6.26) implies the HVZ theorem (if the V ij are continuous andgoing to zero; using the extension mentioned in Remark 1, one can getthe full HVZ result). Given e ∈ S νN − , one defines C ( e ) by putting i and j in the same cluster if and only if e i = e j . It is immediate that theonly right limit in L e is H ( C ( e )). [416] also have an interesting resulton approach to a periodic isospectral torus. Next, I turn to my contributions to the questions of self-adjointnessof Schr¨odinger operators and the more general issue of the proper def-inition of self-adjoint quantum Hamiltonians. In this regard, I shouldmention my work on defining these operators by the method of qua-dratic forms beginning with my PhD. thesis which was published as abook [584]. This thesis studied − ∆ + V on L ( R ) for V ’s obeying(4 π ) − (cid:90) (cid:90) | V ( x ) || V ( y ) || x − y | < ∞ (6.27) a class that I called the Rollnik class , R , after [540]. Since the left side(6.27) is the square of the Hilbert-Schmidt norm of | V | / ( − ∆) − | V | / ,it was rediscovered earlier than my work by many others. In particular,Birman [68] and Schwinger [557] used it in their work (mentioned inSection 8) on bounds on the number of bound states and Grossman-Wu[268] used it in a study of two body scattering theory.The thesis had an interesting source. Wightman was on leave inmy third year of graduate school (1968-69). When he left I didn’thave a definite thesis problem although it seemed possible I’d do athesis on the work I was doing on the anharmonic oscillator. GeorgeTiktopoulos was a High Energy Theory Postdoc at Princeton (latera Professor in Athens) and gave a topics course in potential scatter-ing which, while not mathematically precise, was more mathematicallycareful than many of the other High Energy Theorists. He developedthings for H ≡ − ∆ + V for V ∈ R ∩ L . Such V ’s were not necessarilylocally L so Kato’s theorem didn’t apply and you couldn’t define H as an operator sum. I kept complaining, sometimes being a little ob-noxious as smart graduate students can be, that he needed to add thecondition V ∈ L loc to be able to use Kato’s theorem (Grossman-Wuhad done exactly this). He was insistent that because he could define aGreen’s function for ( H − E ) − for E very negative via a convergent per-turbation series, there must not be a problem. Moreover the physicsshould work for potentials with a | x | − α local singularities so long as α < L loc though requires α < / R works up to 2.I eventually realized that Tiktopoulos was right and one could doeverything using quadratic forms and wrote a long thesis where I rigor-ously pushed through scattering theory through the proof of dispersionrelations and the HVZ theorem. Wightman liked it so much that heproposed making it a volume in the book series he edited for PrincetonPress. I started an instructorship in September, 1969 with the thesislargely written, but Wightman asked me to hold off submission untilhe had a chance to carefully read it and make suggestions. Since I WELVE TALES 53 had a job (in those days, Universities weren’t as picky about postdocswithout being actual docs yet) and he (and I) were busy, submissionkept being postponed. The math and physics departments proposedpromoting me to Assistant Professor and the Dean was very unhappywhen he learned didn’t officially have a degree and refused to processthe appointment until I did. Bob Dicke, the chair in Physics madeit clear to Wightman he’d better deal rapidly with the roadblock andsuddenly within a weekend, he’d read my entire thesis!I wasn’t the first one to use quadratic form methods to define quan-tum Hamiltonians. From the earliest days of quantum mechanics,mathematicians had the idea to use the Friedrichs’ extension whichis essentially a quadratic form construction [680, Section 7.5]. Theperturbation approach that I used had been used in a related contextalready in Kato’s book [363] and by Nelson [483] from whom I learnedit. Forms were also used by several of Nelson’s and Wightman’s stu-dents slightly before me. What I did was show that large parts of thethen existing theory could be carried over to a quadratic form point ofview. Afterwards and, in part, because of my work, forms became atool more widely used by mathematical physics studying NRQM.I returned to form ideas, essentially as a simplifying tool, many othertimes later in my career. On the purely mathematical side, I wrote twopapers on the subject of monotone convergence theorems for forms,an area where the first results appeared in the first edition of Kato’sbook [363]. Both papers [610, 612] resolved an issue left open by Katoin the case of monotone increasing forms; independently, Davies [125],Kato (in the revised second edition of his book) and Robinson [538]had also settled this issue. In [612], I found a decomposition analogousto the Lebesgue decomposition of measures which allowed a significantimprovement of the result for monotone decreasing forms. Also onthe mathematical side, Alonso and I [14] wrote a paper that system-atized the theory of self–adjoint extensions of semi-bounded operatorsin terms of quadratic forms.As noted in Section 2, I (some of joint with Reed) used quadraticform techniques to simplify some of the technical issues around thecomplex scaling results of Balslev-Combes. I developed a quadraticform version of the Cook method in scattering theory [607]; Kato [367],Kuroda [406] and Schechter [551] also had results on that question. AsI’ll discuss soon, in [614], I discussed the form analog of Kato’s famous L loc result.Turning to self-adjointness proper, my most impactful paper was[592]. Following Kato’s work [353], a number of authors studied whatyou needed for essential self-adjointness, aka esa, (on C ∞ ( T ν )) of − ∆ + V on L ( R ν ) in terms of L p ( R ν ) conditions. Call p , ν –canonical if p = 2for ν ≤ p > ν = 4 and p = ν/ p ≥ 5. Then the optimal L p extension of Kato’s theorem is (cid:104) T6.4 (cid:105) Theorem 6.4. Let p be ν –canonical. Then if V ∈ L p ( R ν ) + L ∞ ( R ν ) ,then − ∆ + V is esa on C ∞ ( R ν ) . Except for the improvement that one can have p = ν/ p > ν/ 2) if ν ≥ 5, this is a result of Brownell [82]. For laterpurposes, we note that rather than L p conditions, Stummel [703] statedconditions on V in terms of norms like (when ν ≥ α ↓ (cid:20) sup x (cid:90) | x − y | <α | x − y | − ν | V ( y ) | dy (cid:21) = 0 (6.28) ? ? See the discussion in [117, Section 1.2]. L p conditions imply Stummelconditions but Stummel conditions are more flexible.That Theorem 6.4 is optimal can be seen when ν ≥ 5, then for C large − ∆ − C | x | − is not esa on C ∞ ( T ν ). This implies that Theorem 6.4fails for p larger than the canonical value! In particular, when ν ≥ L potentials for which − ∆ + V is not esa on C ∞ ( T ν ). WhatI realized in [592] is that there is an asymmetry between conditions onthe positive and negative parts of V . In particular, I proved that (cid:104) T6.5 (cid:105) Theorem 6.5. If V ≥ and V ∈ L ( R ν , e − cx d ν x ) for some c > then − ∆ + V is esa on C ∞ ( R ν ) . My discovery (and proof) of this result show the advantage of work-ing in multiple fields, because this was an outgrowth of my work inCQFT! As I discussed in Section 3, for abstract hypercontractive semi-groups (i.e. obeying (3.1) and (3.2)), Segal’s method [558] showed that H + V is esa on D ( H ) ∩ D ( V ) if V obeys (3.3)-(3.4). One of thethings that Høegh-Krohn and I [684] realized is that if (3.3) is replacedby the stronger condition V ≥ 0, one could replace (3.4) by the weakercondition that V ∈ L (we did this to handle the spatially cutoff two di-mensional : exp( ϕ ( x )) : field theory, a favorite model of Høegh-Krohn).I realized that this implied that when (6.29) holds, then for a suitable d , − ∆ + dx + V is esa on C ∞ ( R ν ). An additional trick allowed meto subtract the dx and obtain Theorem 6.5. Given this new result, Imade the natural conjecture for V ∈ L loc ( R ν , d ν x ).When I finished writing the preprint of [592], I mailed a copy to Kato(in those days, papers were typed and, given that Xeroxing was costly,a very few Xerox copies were sent by snail mail - this was years before WELVE TALES 55 TEX and email). About six weeks later (counting the time for groundmail from Princeton to Berkeley and back!), I got Kato’s paper [361]in which he proved (cid:104) T6.6 (cid:105) Theorem 6.6. If V ≥ and V ∈ L loc ( R ν , d ν x ) , then − ∆ + V is esaon C ∞ ( R ν ) . Kato’s method was totally different from mine. He first proved whatI eventually called Kato’s inequality, that∆ | u | ≥ Re(sgn∆ u ) (6.30) (here u is complex valued and sgn( u ) ≡ lim c ↓ ¯ u ( x ) / ( | u ( x ) | + c ) / ).A novel feature of (6.30) is that Kato proved it as a distributionalinequality under the conditions that u ∈ L ( R ν ) and ∆ u ∈ L ( R ν ).This inequality came from left field - I’m not aware of anything closeto it in the work of Kato or anyone else. Moreover, given the inequality,the proof of Theorem 6.6 is a few lines (see, for example, [680, pg 622]).Kato’s paper did much more than just prove (6.30) and show howto use that to prove Theorem 6.6. He also proved a version of (6.30)for magnetic fields, namely for smooth −→ a ∆ | u | ≥ Re(sgn( −→∇ − i −→ a ) u ) (6.31) again as a distributional inequality. He used this to show that H ( −→ a , V ) = − ( −→∇ − i −→ a ) + V (6.32) is esa on C ∞ ( R ν ) if V ∈ L loc ( R ν ) , V ≥ −→ a is C . I then improvedthis [593] to only require that div( −→ a ) ∈ L loc ( R ν ) and −→ a ∈ L ploc ( T ν )where p had a slightly stronger condition than 2 p being ν canonical.Finally, Leinfelder-Simader [426] proved the optimal result requiringthat div( −→ a ) ∈ L loc ( R ν ) and −→ a ∈ L loc ( R ν ).Kato also had results where V had a negative part. In that context,he introduced what I later called the Kato class, K ν . V ∈ K ν ⇐⇒ lim α ↓ (cid:104) sup x (cid:82) | x − y |≤ α | x − y | − ν | V ( y ) | d ν y (cid:105) = 0 , if ν > α ↓ (cid:104) sup x (cid:82) | x − y |≤ α log( | x − y | − ) | V ( y ) | d ν y (cid:105) = 0 , if ν = 2sup x (cid:82) | x − y |≤ | V ( y ) | dy < ∞ , if ν = 1(6.33) ? ? Ironically, this is not optimal (but it is close to optimal) forself-adjointness, but it is optimal for various L p semigroup condi-tion as shown by Aizenman-Simon [12, Theorem 1.3], for exam-ple V ∈ K ν ⇐⇒ e − t ( − ∆ −| V | ) is bounded from L ∞ to L ∞ withlim t ↓ (cid:107) e − t ( − ∆ −| V | ) (cid:107) ∞ , ∞ = 1. The Kato class will appear several timesbelow. Before leaving the subject of Kato’s inequality and self-adjointness,I note the following form analog of Theorem 6.6 that I proved usingconnections of L p semigroup bounds and Kato’s inequality that I’lldiscuss shortly. ? (cid:104) T6.7 (cid:105) ? Theorem 6.7 (Simon [614]) . Let V ≥ be in L loc ( R ν , d ν x ) andlet −→ a ∈ L loc ( R ν , d ν x ) be an R ν valued function. Let Q ( D j ) = { ϕ ∈ L ( R ν , d ν x ) | ( ∇ j − ia j ) ϕ ∈ L ( R ν , d ν x ) } with quadratic form (cid:104) ϕ, − D j ϕ (cid:105) = (cid:107) ( ∇ j − ia j ) ϕ (cid:107) . Let h be the closed form sum (cid:80) νj =1 − D j + V . Then C ∞ ( R ν ) is a form core for h . This paper also has a proof of Theorem 6.6 using L p semigroupbounds. It doesn’t explicitly use Kato’s inequality but, by then, Iknew that his inequlaity was a expression of the positivity preservingbehavior of semigroups.Both Kato and I were taken with his inequality and each of us wroteadditional papers on the subject (Kato [362, 365, 366, 368, 81, 369,370]). I focused on what the analog is in a much more general context.In [609], I proved ? (cid:104) T6.8 (cid:105) ? Theorem 6.8 (Simon [609]) . Let A be a positive self–adjoint operatoron L ( M, dµ ) for a σ –finite, separable measure space ( M, Σ , dµ ) . Thenthe following are equivalent:(a) ( e − tA is positivity preserving) ∀ u ∈ L , u ≥ , t ≥ ⇒ e − tA u ≥ (b) (Beurling–Deny criterion) u ∈ Q ( A ) ⇒ | u | ∈ Q ( A ) and q A ( | u | ) ≤ q A ( u ) (6.34) ? ? (c) (Abstract Kato Inequality) u ∈ D ( A ) ⇒ | u | ∈ Q ( A ) and for all ϕ ∈ Q ( A ) with ϕ ≥ , one has that (cid:104) A / ϕ, A / | u |(cid:105) ≥ Re (cid:104) ϕ, sgn( u ) Au (cid:105) (6.35) ? ? The equivalence of (a) and (b) for M a finite set (so A is a matrix)is due to Beurling–Deny [67]. For a proof of the full theorem (whichis not hard), see Simon [609] or [680, Theorem 7.6.4]. In that paper,I also conjectured the analog of (6.31) in a similar general context, aresult then proven independently by me [618] and by a group of threeothers [295] (cid:104) T6.9 (cid:105) Theorem 6.9 (Hess–Schrader–Uhlenbrock [295], Simon [618]) . Let A and B be two positive self–adjoint operators on L ( M, dµ ) where ( M, Σ , dµ ) is a σ –finite, separable measure space. Suppose that ϕ ≥ ⇒ e − tA ϕ ≥ . Then the following are equivalent: WELVE TALES 57 (a) For all ϕ ∈ L and all t ≥ , we have that | e − tB ϕ | ≤ e − tA | ϕ | (b) ψ ∈ D ( B ) ⇒ | ψ | ∈ Q ( A ) and for all ϕ ∈ Q ( A ) with ϕ ≥ andall ψ ∈ D ( B ) we have that (cid:104) A / ϕ, A / | ψ |(cid:105) ≤ Re (cid:104) ϕ, sgn( ψ )B ψ (cid:105) (6.36) ? ? For a proof, see the original papers or [680, Theorem 7.6.7]. There isfurther discussion of this result in the context of diamagnetic inequal-ities in quantum mechanics in Section 7.These semigroup ideas are intimately related to properties of eigen-functions of Schr¨odinger operators, a subject I often looked at in the1970’s. One issue that particularly attracted me was that of exponen-tial decay. In 1969, the only results on decay of discrete eigenfunctionsof N -body quantum Hamiltonians with N > N = 3 or only Coulomb potentials. I gave looking at N –body systems to Tony O’Connor, my first graduate student (whobegan working with me when I was a first year instructor). He had theidea of looking at analyticity of the Fourier transform and obtainedresults in the L sense (i.e. e a | x | ψ ∈ L ) that were optimal in thatyou couldn’t do better in terms of isotropic decay. Here | x | is a massweighted measure of the spread of the N particles, explicitly, in termsof the inner product (6.5) and the center of mass ρ N of (6.6), one hasthat M | x | = (cid:104) r − ρ N , r − ρ N (cid:105) (6.37) O’Connor found one had the L bound if | a | < M (Σ − E ).His paper [490] motivated Combes–Thomas [108] to an approachthat has now become standard of using boost analyticity. It is widelyapplicable although in the N -body case it exactly recovered O’Connor’sbound. Over the years, I had a six paper series on the subject ofexponential decay [596, 597, 598, 131, 93, 444]. In the first three papers,I looked at getting pointwise bounds. In the first paper, I obtainedoptimal pointwise isotropic bounds for N –body systems. In the secondpaper, I considered the case where V goes to infinity at infinity andproved pointwise exponential decay by every exponential (Sch’nol [553]earlier had a related result). In the third paper, I assumed | x | m lowerbounds and got exp( −| x | m +1 ) pointwise upper bounds. When one hasan upper bound on V of this form, one gets lower bounds of the sameform on the ground state. Papers 1-2 were written during my fall1972 visit to IHES, one of my most productive times when Lieb and Idid most of the Thomas–Fermi work and I developed new aspects ofcorrelation inequalities and Lee–Yang for EQFT. The fourth paper [131] (joint with Deift, Hunziker and Vock; Deifthad been my student and we continued working on this while he was apostdoc. I learned that Hunziker was looking at similar questions so wejoined forces – Vock was his master’s student) explored non–isotropicbound for N -body systems. We found a critical differential inequal-ity that if f obeys it, then e f ψ ∈ L ∞ and in some cases were able tofind explicit formula for the optimal f (but only in a few simple situa-tions). Later, Agmon [4] found the optimal solution of the differentialinequality as a geodesic distance in a suitable Riemann metric (discon-tinuous in the case of N –body systems) – this is now called the Agmonmetric, a name that appeared first in the fifth paper of this series byCarmona–Simon [93], which also proved lower bounds for the groundstate complementary to Agmon’s upper bounds. We proved that if ψ ( x ) is the ground state and ρ ( x ) the Agmon metric distance from x to 0, then lim | x |→∞ − log | ψ ( x ) | /ρ ( x ) = 1. In some ways, the fourthpaper is made obsolete by [4, 93] although the explicit closed form for ρ in some cases remains of significance. The sixth paper with Lieb[444] studied N -body system in the special region where subclustersremained bound but were distant from each other.Carmona–Simon [93] used path integral techniques in NRQM so Ipause to say something about that subject which due to pioneeringwork of Lieb and Nelson was a kind secret weapon around Princetonwhich I also used so extensively that I wrote what became a standardreference [661] based on lectures I gave in Switzerland in the summerof 1977. I was on leave in 1976-77 and also gave lectures at the Uni-versity of Texas which also turned into a book [662] on my other secretfunctional analytic weapon, the theory of trace ideals. It has had arebirth of use since it is a tool in quantum information theory.One thing that I used path integral methods for is to study moregeneral issues of properties of eigenfunctions and integral kernels (forthe semigroup and resolvent) than exponential decay, although theyalso allowed stronger results and simpler proofs for exponential de-cay. I did this in the Functional Integration book just mentioned buteven more in two articles, one with Aizenman [12] and one that wasbilled as a review article [626]. The article with Aizenman, whichwon the Stampachia prize, proved Harnack inequalities and subsolu-tion estimates on eigenfunctions of Schr¨odinger operators under only K locν conditions on V (see also [106]). The 80 page review article is myfifth most cited publication (the only more cited items are three booksand the Berry’s phase paper) and proves many results, for example,on continuum eigenfunction expansions, under greater generality than WELVE TALES 59 previously. I should mention that this work was influenced by a beau-tiful paper of Carmona [87] (and later [88]) that emphasized a simpleway to get L ∞ bounds. When I learned of this work, I invited Ren´eto visit Princeton leading to [93]. Later, he and I teamed up with anIrvine graduate student of his to discuss analogs of these Schr¨odingeroperators results when − ∆ is replaced by other generators of positivitypreserving semigroups, most notably the one, √− ∆ + m , associatedto relativistic quantum theory [92].A little more on subsolution estimates (which had been discussed formore general elliptic operators but with greater restrictions on the reg-ularity of coefficients in the PDE literature, especially by Trudinger).These imply that if Hu = Eu , then | u ( x ) | ≤ C (cid:90) | y − x |≤ | u ( y ) | d ν y (6.38) ? ? where C only depends on K ν norms of V restricted to the ball of radius1 about x. These immediately imply that L estimates, for examplethose found by O’Connor [490], imply pointwise estimates, for example,those proven by me in [596].Eigenfunction properties and expansions recurred in my later workmany connected with 1D problems, including the discrete case, espe-cially with applications to a.c. spectrum and/or almost periodic prob-lems (see Section 9). I mention four: [654] has a simple proof that,for 1D discrete and continuum Schr¨odinger operators, if all eigensolu-tions are bounded for energies in an set, S , then the spectrum is purelya.c. on S , a result of Gilbert-Pearson [233]. It also uses this theo-rem to analyze 1D Schr¨odinger operators with potentials of boundedvariation, recovering results of Weidmann [729] and extending them tothe Jacobi case. [663] gave a simple proof, using rank one perturba-tion theory (specifically Theorem 11.2(b)), that for such operators on L ( R ), the singular spectrum is always simple, a result proven earlierby Kac [347] and Gilbert [232] in a more complicated way. Last-Simon[415] has many results on the connection of eigenfunctions to spectralbehavior depending on the growth of transfer matrices for ODEs andKiselev-Last-Simon [383] has additional results on growth of transfermatrices and spectral properties, including the subtle borderline x − / decaying random potential.Before leaving the subject of eigensolutions, I should mention a paperwith Schechter [552] also written the 1975-76 year that I was on leave(Schechter was at Yeshiva University where I spent two days a weekthat year). Carleman [86] had studied the issue of unique continuation (if a solution of Hu = Eu vanishes on an open set, it vanishes iden-tically), a subject for which almost all work since has used what havecome to be called Carleman estimates after that paper. In 1959, Kato[358] understood that unique continuation was an element of a proofof the non-existence of positive eigenvalues. What Schechter and I re-alized is that Carleman estimates were limited to bounded potentialsand it was natural to consider the problem for some unbounded V ’s.We proved the first such results although we stated that we believedour conditions were far from optimal. We hoped that we’d stimulatethe harmonic analysis community and there were a number of papersthat our work stimulated. Most notable were Jerison-Kenig [336] andKoch-Tataru [391]. An optimal result (from [336]) says that one hasunique continuation for − ∆+ V is V ∈ L ν/ loc ( R ν ) (with ν > L p conditions this is optimal but it has been realized recently(Garrigue [212]) that if ν = N µ and V has an N body form it is notoptimal. One would hope that there is a result for L ploc when p > µ/ p > ν/ N very large. Ironically, theresult of Schechter-Simon [552] that p > µ (if µ > 4) suffices is amongthe strongest results for this general N -body case. In any event, thereis work remaining to be done.I’d first heard of unique continuation theorems as a graduate studentin the context of Kato’s result [358] that if V is a continuous functionon R ν so that | x || V ( x ) | → − ∆+ V has no eigenvaluesin (0 , ∞ ). In one of my first serious papers [579], I found a result onno positive eigenvalues that allows V to be a sum of two pieces, V that obeys Kato’s condition and a piece, V , that obeys V → | x | ∂V /∂x → V going to 0 at ∞ so that − ∆ + V has a positive eigenvalue. The example they actually write down has V ( x ) = O( | x | − ) at infinity and violates Kato’s theorem! I discoveredthat they had clearly used cos x/ sin x = tan x which caused a miracu-lous cancellation of the O( | x | − )! My paper seems to have been the firstto note the error and write down the correct explicit form they shouldhave. At one point, I had to ask Wigner a question about somethingelse and I asked him about if he knew that this paper, written 40 years,before had this error. He thought for a moment and then replied “No,I didn’t know” and, after a pause “Johnny did that calculation.”Kato’s result says that on (0 , ∞ ), if lim x →∞ x | V ( x ) | = 0, then h = − d dx + V ( x ) has no eigenvalues in (0 , ∞ ) and many years later Kiselev, WELVE TALES 61 Last and I [383] proved that if lim x →∞ x | V ( x ) | < ∞ , the set of positiveeigenvalues is discrete with only 0 as a possible limit point (indeed,the sum of the positive eigenvalues, if any, must be finite). In [655], Iconstructed V ’s where x | V ( x ) | had arbitrarily slow growth at infinity(in particular, some for which it was known that the a.c. spectrum was[0 , ∞ ) by Kiselev [381]) with any desired uncountable set of positiveenergy eigenvalues, even dense sets. I was motivated by an earlierpaper of Naboko [478] who was able to construct such examples solong as the set of positive eigenvalues had the form E n = κ n with the κ n rationally independent. Hsu et al. [307] have a recent review ofphysically relevant examples with bound states in the continuum.Next, we turn to scattering theory, in particular the question of N-body asymptotic completeness (big problem 4) where my most signifi-cant result involves the Deift-Simon wave operators [134]. To put it incontext, I begin with a lightening summary of the high points of 2 and N body scattering. One needs to bear in mind that big problem 3 (ab-sence of s.c. spectrum) is often intimately related to big problem 4 inthat sometimes the techniques to solve them (namely detailed analysisof the boundary values of the resolvent) are close; indeed Reed-Simon[527] calls the combination of the two, strong asymptotic completeness .In abstract scattering theory, one defines wave operators for a pairof self-adjoint operators by (as above, the funny ± convention is takenfrom the physics literature and often the opposite to the convention inthe mathematics literature).Ω ± ( A, B ) = lim t →∓∞ e itA e − itB P ac ( B ) (6.39) where P ac ( B ) is the projection onto those the subspace of those vectorswhose spectral measure for B is purely absolutely continuous. Theinsertion of the P ac ( B ) (which is redundant in the usual case of twobody physics where B = − ∆ , A = − ∆ + V with V short range) isa wonderful realization of Kato [356], who understood that if we callΩ ± ( A, B ) complete if and only ifran Ω ± ( A, B ) = H ac ( A ) ≡ ran P ac ( A ) (6.40) then one has that (cid:104) T6.10 (cid:105) Theorem 6.10. Suppose that Ω ± ( A, B ) exist. Then, they are completeif and only if Ω ± ( B, A ) exist. The first mathematical results on existence of wave operators was asimple argument of Cook [110], improved by Hack [276] and Kuroda[405]. The latter two got existence on R ν for V ’s decaying as | x | − − ε . The first completeness results were obtained by Kato [356, 357] andRosenblum [541] whose best result says that if A − B is trace class, thenΩ ± ( A, B ) exist so, by symmetry and Theorem 6.10, they are complete(see [682, Section 13] for a lot more on the history and extension of theKato-Rosenblum theorem). By shifting from trace class to differencesof resolvent being trace class, these results imply completeness on R if V decays like | x | − − ε . It then took about ten years, to get to solvingbig problems 3 and 4 for N = 2 and V bounded by | x | − − ε . This wasfirst accomplished by Agmon [3] (Agmon announced this at the 1970ICM; at the same conference, Kato [364] announced the solution of bigproblem 4 for this class, extending some ideas of Kato-Kuroda. ThenKuroda [407, 408] realized that by borrowing one technical device fromAgmon, their method also solved big problem 3 for this class. For moreon the history and details of this work see [682, Section 15]).Already in 1963, Faddeev [179] obtained asymptotic completenessfor certain 3 body equations. Because his basic condition were writtenon the Fourier transform, it is difficult to write them in terms of the V ij but his assumptions required decay faster than (1 + | x | ) − − ε . He alsosupposed the two body subsystems didn’t have zero energy resonances.In any event, his work had limited impact on the mathematical physicsliterature and his methods were never extended to N ≥ N -body systems. During that period, itwas a major open question and several people, formally or informally,announced solutions which turned out to have errors. At one point,Agmon wryly remarked to me “those whom the gods would drive mad,they teach of the problem of N -body asymptotic completeness.” WhileI was certainly aware of the problem and several times did work relatedto it, I never tried to systematically approach it because I didn’t seea fruitful approach. Two high points of the fifteen year intermediateperiod were work of Enss and Mourre, each of which played impor-tant roles in the eventual resolutions. Because Mourre’s work is moreconnected with big problem 3, I’ll postpone its discussion.Enss [172] revolutionized scattering, especially two body scattering.At a heuristic level, scattering is a time-dependent phenomenon butprior to Enss, the most powerful results in quantum scattering usedtime independent methods (i.e. focus on resolvents rather than theunitary groups) - Faddeev’s work and the Agmon-Kato-Kuroda workmentioned above. Enss used purely time-dependent methods without WELVE TALES 63 any resolvents anywhere. He combined Cook’s method with two extraingredients. The first was geometric, motivated, in part, my work withDeift [134], discussed below, and the geometric approach to the HVZtheorem by Enss [171] and me [608], discussed above. The other wasto localize in phase space. He did this while respecting the uncertaintyprinciple, by, in essence, projecting on spectral subspaces for the dila-tion operator, A = [ x · p + p · x ]. This suggested a natural way ofapproaching his work was to use an eigenfunction expansion for A , i.e.the Mellin transform, which is precisely the approach used by Perry[506] in a thesis done under my direction.I was taken with this work of Enss and talked it up using my thenconsiderable influence. I wrote a long (50 page) article [616] showinghow to apply it in a large number of scattering theory situations (Reedand I had just finished our scattering theory volume [527], so I knew oflots of scattering situation beyond 2-body NRQM). When Enss visitedthe Institute, we used some of these ideas to study total cross-sections[173, 174].Fifteen years after Enss’ work, his techniques was critical to an anal-ysis by me and others of some intriguing examples of Neumann Lapla-cians. Recall that Dirichlet and Neumann Laplacians are describedmost naturally in terms of quadratic forms [680, Section 7.5]. Givenan open set, Ω ⊂ R ν , one defines Q ( − ∆ Ω N ) to be the set of func-tions, ψ ∈ L (Ω , d ν x ) so that ∇ ψ ∈ L (Ω , d ν x ). The sesquilinear form ψ (cid:55)→ (cid:82) |∇ ψ | d ν x , where ∇ ψ is the distributional gradient, defines aself adjoint operator, − ∆ Ω N , the Neumann BC Laplacian for Ω. If werestrict the form to the closure of C ∞ (Ω) in form norm, the corre-sponding operator is − ∆ Ω D , the Dirichlet BC Laplacian for Ω. If Ωhas a smooth boundary, the functions C ∞ with Dirichlet or Neumannboundary conditions are an operator core for − ∆ Ω D and − ∆ Ω N respec-tively.As I’ll explain in Section 8, I had considered the case Ω = { ( x, y ) | | xy | < } ⊂ R and had shown that despite the fact thatΩ has infinite volume, − ∆ Ω D has discrete spectrum. The intuition isthat the horns got narrow so the lowest Dirichlet eigenvalue associatedwith cross sections went to infinity. When chatting at a conference inGregynog, Wales, Brian Davies and I considered − ∆ Ω N . With NeumannBC, the lowest eigenvalue is zero so one expects one ac mode in each ofthe four horns. It was easy to construct wave operators to get existenceand we realized that using Enss theory, we could prove [128] that − ∆ Ω N had ac spectrum of multiplicity exactly four (one for each horn). The opposite phenomenon to Dirichlet Laplacians of infinite volumebut discrete spectrum is Neumann Laplacians with finite volume andsome essential spectrum. Already Courant-Hilbert [112] had foundbounded regions with 0 ∈ σ ess ( − ∆ Ω N ) and Hempel, Seco and I [287]had shown that for any closed set S ⊂ [0 , ∞ ), there was a boundedΩ so that σ ess ( − ∆ Ω N ) = S ; these examples had empty ac spectrum.Davies and I found that for Ω = { ( x, y ) | x > , | y | < f ( x ) } , one hasthat − ∆ Ω N has ac spectrum of multiplicity one on all of [0 , ∞ ) so longas all of k ( x ) = | f (cid:48) ( x ) | , k ( x ) = | f (cid:48) ( x ) | f ( x ) − and V ( x ) = 14 (cid:18) f (cid:48) f (cid:19) + 12 (cid:18) f (cid:48) f (cid:19) (cid:48) (6.41) are O( | x | − − ε ) as x → ∞ . (In Section 8, I’ll discuss results of [331]on eigenvalue asymptotics when V ( x ) → ∞ ). For example if f ( x ) = x − α , α > 0, one gets ac spectrum even though if α > 1, then Ω hasfinite volume. I realized [645] that one could wrap such a finite volumehorn up and construct a bounded region whose − ∆ Ω N had ac spectrumon [0 , ∞ )!Returning to the theme of general N -body asymptotic completeness,the big breakthrough and first solution was by Sigal-Soffer [576]. Thepaper uses in impressive combination of Mourre estimates, Deift-Simonwave operators and phase space estimates motivated by Enss theory.Unfortunately, as the MathSciNet review says: It is disappointing thatthis important result has not received the exposition it deserves. Thepaper contains numerous misprints, points of unclarity, obscure nota-tion, and minor technical errors. Because the details of the proof areinaccessible to all but the most dedicated specialist, there has been con-siderable speculation about the validity of the result. Fortunately, twoexperts who have studied the paper thoroughly have assured the reviewer(open letter from W. Hunziker and B. Simon, dated September 1, 1987)that essentials of the proof are correct. I should say a little more aboutmy roll in this. The 70+ page paper was clearly very important butalso not ideally written. Walter Hunziker and I felt a duty as leadingfigures (and also, because, if I recall, one or both of us were refereesfor the Annals) to determine if the results were correct. I visited ETHin the summer of 1986 and for three weeks, I arrived about 10 in themorning, worked with Walter until 3 in the afternoon (with a break forlunch) painfully plowing through the paper line by line. We found lotsof little errors - typically a lemma was wrong but when we figured outhow it was used, the lemma and/or its proof could be slightly modifiedto work. The authors had apparently decided to state all lemmas inthe most general form they could, often so general, it was now wrong. WELVE TALES 65 We made numerous suggestions for changes, decided the paper was ba-sically correct and recommended publication even though we agreedthat even after changes it was not a model of exposition. When theMath Reviews reviewer contacted me, Walter and I produced a publicdocument vouching for the result.Later proofs sharing some elements with [576], are due to Graf [251],Derezi´nski [153] and Yafaev [739]. Coulomb and long range potentialsare treated in [577, 153].Having put it in context with this summary, I turn to exactly whatDeift and I did in [134], a paper used in all the proofs mentionedabove [576, 577, 251, 153, 739]. We proved a kind of N -body analogof Theorem 6.10. For any partition, C , we defined | r | C in (6.24) as theminimal distance between components. We also can define ρ C ( r ) = max ( jq ) ⊂C | r j − r q | (6.42) ? ? which describes how far particles within the components are from eachother. One defines functions ˜ J C for each C ∈ P which are 1 in the regionwhere | r | > ρ C ( r ) ≤ [ | r | C ] / (6.43) and supported in the union of the set where | r | ≤ | r | is large,points in the support have distances within the clusters much less thanthe average distance between particles. Given two partitions C and D ,we write D (cid:3) C if D is a refinement of C , i.e. if every subset in C is aunion of sets in D . Deift and Simon define J C = ˜ J C − (cid:88) D (cid:3) C , D(cid:54) = C ˜ J C (6.44) ? ? which eliminates configurations where the particles in subsets of D arevery close to each other while particles in different subsets of D , butin the same subset of C , are quite far but still small compared to themaximum intercluster distance. The Deift-Simon wave operators aredefined by Ω ±C = s-lim t →∓∞ e itH ( C ) J C e − itH P ac ( H ) (6.45) ? ? If this limit exists and, say ψ = Ω + C ϕ , for ϕ ∈ H ac ( H ( C )) then as t → −∞ , e − itH ψ is asymptotic to J C e − itH ( C ) ϕ . If all the component H ( C j )’s have no sc spectrum and obey asymptotic completeness, then J C e − itH ( C ) ϕ will look exactly like a sum of e − itH η t for t a thresholdassociated to C and suitable η t ∈ ranΩ + t . Thus one expects, and Deift-Simon prove, an analog of Theorem 6.10 ? (cid:104) T6.11 (cid:105) ? Theorem 6.11 (Deift-Simon [134]) . Suppose H is an N body quan-tum Hamiltonian so that all the proper subset Hamiltonians H ( C ) haveno singular spectrum and obey asymptotic completeness. Then H isasymptotically complete if and only if all the wave operators Ω ±C exist. This clearly suggests an approach to proving asymptotic complete-ness inductively.I should say something about the origin of this work with Percy Deift.Percy, who is actually a year older than me, grew up in South Africaand, encouraged to study practical subjects, got a master’s degree therein Chemical Engineering. When he got interested in more theoreticalthings, he decided to go Princeton, applying to a strange program inapplied math. At the time Princeton had no Applied Math Dept. Mar-tin Kruskal, because of his famous work on the Schwarzschild solution,had an appointment in Astronomy, but his true love was mathematics(later, once his soliton work became famous, I persuaded my colleaguesin mathematics to give him a joint appointment, but that is anotherstory) . He convinced the administration to allow him to run a PhDprogram where students had to have a home department although theycould do their thesis work in any area and their preliminary exams werein applied math with various faculty in other departments on theirexam committee. It took students who often wouldn’t have been ableto get into the departments in which they did their thesis work by thefront door. It’s students were a mixed bag. Some of the most painfulqualifying exams I ever served on were in that program, students whowere woefully prepared and didn’t come close to passing. On the otherhand, it had some spectacular successes. Ed Witten, who would neverhave been admitted to the math or physics departments because he’dhad almost no courses in those subjects when he applied, did get intoKruskal’s program (one of the decisions I made as director of graduatestudies in Physics was, after getting rave reviews from some professorswho had him in courses, to allow him to transfer to the regular physicstrack). Percy Deift, who has been my most successful student, wasin Kruskal’s program, originally with Chemical Engineering as a homedepartment.I was on leave in Percy’s first year in Princeton and he came to mein the middle of his second year saying that he wanted to do a thesisunder by direction. I was skeptical but after consulting Lon Rosen,with whom he’d taken a math methods course, I said I’d give him atry. We discussed various open problems in the then several areas Iwas working on. He came back and told me that he’d like to work on WELVE TALES 67 N -body asymptotic completeness. I told him that was a totally crazyproblem for a graduate student to work on without some wonderfulidea that looked almost sure to lead to success – it was too risky thatit would lead to total failure after several years. He went away and cameback and said that he understood it was too hard but could I suggestsome problems that would lead in the direction of eventually solving N -body asymptotic completeness. His approach made me decide thatmaybe this student might get somewhere after all. In the end, his nicethesis was in another direction [129] but we agreed to discuss scatteringtheory on the side. This was before the work that Enss and I did onHVZ or Enss’ work and we wound up discovering the usefulness ofusing geometric ideas (indeed [134] was motivation for my geometricapproach to HVZ [608]). We first wrote a cute paper [133] that usedtrace class methods, Dirichlet decoupling and path integral techniquesto prove that (positive) local singularities were irrelevant to questionsof existence and completeness of two body Schr¨odinger operator waveoperators and our second paper was [134]. (We wrote several otherpapers together later on in his career).That completes what I want to say about big problem 4 so I turn tobig problem 3 and the work I did with Perry and Sigal on extendingMourre theory to N -body systems [507]. Around 1977, Eric Mourrewrote a preprint and submitted it to one of the then editors of Com-munications in Mathematical Physics handling Schr¨odinger operators(starting two years after that, I served as the main editor for that sub-ject for more than 35 years). Then, as now, an editor has the freedomto look over a manuscript and reject it without any refereeing and theeditor decided that Mourre’s paper wasn’t important enough for CMPand rejected it. Mourre placed the manuscript in his desk drawer ratherthan submit it elsewhere. I also got the preprint, thought it might beinteresting, but wasn’t sure since it was hard to follow. I was short ontime, so I passed copies on to two of my former students asking themto take a careful look and let me know if there was anything interestingthere. They each eventually reported there didn’t seem to be anythingworth spending a lot of time on. Despite these initial opinions, thepaper was one of the most significant in the study of N -body NRQMspectral theory!Part of Mourre’s motivation was work by Lavine [418, 419, 420, 421]on N body systems with repulsive potentials who, in turn, was extend-ing work of Putnam [516], Kato [360], Weidmann [728] and Kalf [348].Lavine and Mourre centrally use the generator of dilations (which isalso central, in a different way to the work of Enss and Perry as I’ve discussed): A = 12 i ( x · ∇ + ∇ · x ) (6.46) What makes repulsive potentials special is that one has a positive com-mutator [ H, iA ] > where H is the N -body Hamiltonian. Putnam (with a later slick proofof Kato) proved that for bounded operators, positive commutators im-plies both have purely a.c. spectrum (related to this and useful inMourre theory is the Virial theorem which is where the work of Wei-dmann and Kalf comes in). Lavine figured out how to modify thingsto apply to the unbounded operators that enter in (6.47) and also howto sometimes get scattering theory results using an extension of Kato’ssmoothness theory [359, 360].In his paper, what Mourre realized is that one could get a lot froma local version of (6.47). Namely, he considered what have come to becalled Mourre estimates P ∆ ( H )[ H, iA ] P ∆ ( H ) ≥ αP ∆ ( H ) + K (6.48) where ∆ = ( a, b ) is an open interval, P ∆ a spectral projection, α > K a compact operator. First of all, Mourre showed that when(6.48) holds, H has no s.c. spectrum on ∆ and that H has only finitelymany embedded eigenvalues in any [ a + δ, b − δ ] , δ > 0, each of finitemultiplicity. Secondly, he showed that (6.48) held for fairly general3-body Schr¨odinger Hamiltonians with decaying potentials for ∆ anyinterval avoiding 0 and the two body thresholds (which means eigen-values of two body subsystems).In one sense, it is surprising that Mourre’s path breaking work wasn’trapidly recognized but there are several reasons. First the paper wasin French (the CMP editor was French so that wasn’t an issue on therapid initial rejection). It was partly the originality of many of theideas. The estimate (6.48) with the compact error and its proof viadifferential inequalities were so new it was a little difficult to graspwhat was going on. But most of all, the paper was terribly written. Itwas often unclear what the author was doing and what his steps meant.Fortunately, the paper didn’t merely wind up in his desk drawerbecause Mourre gave talks on it at conferences and, at one, Israel Sigalrealized there might be something important here (and indeed Mourreestimates were a critical step in Sigal’s 15 year successful quest to prove N -body asymptotic completeness). So Sigal decided to try and extendMourre’s work to N > 3. The first problem he faced is that the preprintwas in French and he didn’t speak the language. But he learned that WELVE TALES 69 Peter Perry, who was then my graduate student, was fluent in French,so they started working together and when trying to overcome sometechnical issues, they decided to ask me to join them. By exploitingsome rather complicated arguments, we were able to prove ? (cid:104) T6.12 (cid:105) ? Theorem 6.12 (Perry-Sigal-Simon [507]) . Let H be a reduced N -bodyHamiltonian with two body potentials obeying: | V ij ( x ) | ≤ C (1 + | x | ) − − ε (6.49) for some C, ε > . Then the closure of the set of thresholds is count-able, and a Mourre estimate of the form (6.48) with A given by (6.46) holds for any closed interval in the complement of the closure of thethresholds. This implies that H has no singular continuous spectrumand that any such closed interval has at most finitely many eigenvalues,each of finite multiplicity. Remarks. 1. (6.49) is stated for simplicity of exposition; local singu-larities are allowed and one can even have slower than | x | − decay ifthe first or second derivative decays as in (6.49).2. We first of all showed (following Mourre) that Mourre estimatesimplies absence of sc spectrum and finiteness of embedded eigenvalues.One then gets the result inductively on thresholds so the key is theproof of the Mourre estimates which is quite involved in our paper.3. Prior to our work, with one exception all results on absence ofsc spectrum were single channel, requiring either repulsive potentials[418, 419, 420, 421] or weak coupling [322]. The exception is the resultsof Balslev-Combes [52] using dilation analyticity that required analyticpotentials. Our work was the first that, for example, handled C ∞ potentials.This solved big problem 3 in great generality and provided toolsof use in scattering theory. After we obtained our results, I contactedMourre to find out where his paper had been published. When I learnedit was sitting in his drawer, with his permission I contacted the orig-inal rejecting editor and the editor-in-chief of CMP (where I was, bythen, an editor) to get their OK to have the paper reconsidered. Pe-ter Perry (with Mourre’s permission) translated the paper into Englishand it appeared as [477]. Before leaving the subject of Mourre theory,I should mention two lovely papers of Froese-Herbst and one paperfor which I was a coauthor. In [196], Froese-Herbst found a consider-ably streamlined proof of N -body Mourre estimates and, in [197], theyused Mourre estimates to obtain some remarkable results on exponen-tial decay. In [65], Bentosela-Carmona-Duclos-Simon-Souillard-Wederused Mourre theory (but with A = i∂/∂x which works well because of the F x term) to prove that a Stark Hamiltonian, − ∆ + V ( x ) + F x ,with V fairly smooth and F (cid:54) = 0, has no sc spectrum and only iso-lated point spectrum of finite multiplicity (in 1D, a separate argumentproved no eigenvalues).That completes what we will say about few-body quantum systems.Since this is the main section that includes a discussion of scatteringtheory, we use the rest of this section to discuss some other work ofmine connected scattering starting with inverse scattering. My mostimportant inverse result involves an alternative to the Gel’fand-Levitan[214] approach to determining the potential, V , for h = − d dx + V on [0 , ∞ ) with Dirichlet boundary conditions at 0 from its spectralmeasure (in the physics literature this is usually discussed in terms ofdetermining V from a reflection coefficient, bound state energies andnorming constants but they determine the spectral measure, i.e. themeasure dρ with (cid:90) f ( x ) dρ ( x ) = π − lim ε ↓ (cid:90) f ( x ) (cid:104) δ (cid:48) , ( h − x − iε ) − δ (cid:48) (cid:105) dx (6.50) ? ? and it is the process ρ (cid:55)→ V that concerned Gel’fand-Levitan). I wrotethree papers on this alternate approach (the second and third with,respectively, Gesztesy and Ramm) [657, 225, 519] and a fourth withGesztesy [227] applying it..To understand my motivation, one needs to understand the discreteanalog of this question - going from a probability measure of boundedsupport on R to the Jacobi parameters, { a j , b j } ∞ j =1 where each a j isstrictly positive and each b j is real (see [674, Section 1.2] for discussionof Jacobi parameters). The simplest way is as recursion coefficients fororthogonal polynomials on the real line (aka OPRL): given, dµ , oneforms the orthonormal polynomials and finds the recursion parameters[674, (1.2.15)]. But there is another method associated with 19 th cen-tury work of Jacobi, Markov and Stieltjes. If dµ is a measure on R ofcompact support, one defines the m -function by: m ( z ) = (cid:90) dµ ( x ) x − z (6.51) The operator of multiplication by x in L ( R , dµ ) represented in theOPRL basis yields a tridiagonal matrix with b n on diagonal and a n offdiagonal called a Jacobi matrix . Mark Kac once gave talks around thetopic he described as “be wise, discretize”. Indeed, around 1980, therewas a notable shift in my work and the work of many mathematicalphysicists (and even earlier in some of the condensed matter theoretical WELVE TALES 71 physics literature) toward difference rather than differential operatorswhich are often technically simpler. On the half line, this was often discrete Schr¨odinger operators which are Jacobi matrices with a n allequal to 1. One also studies its analogs on all of Z and on Z ν andthe more general Jacobi case. Around 2000, when I added orthogonalpolynomials on unit circle (aka OPUC) to my repertoire (see [664, 665])the spectral theory of the associated operator theory and its matrixrepresentation (the CMV matrix discussed in [664, Chapter 4]) alsobecame an interest.If dµ has Jacobi parameters { a j , b j } ∞ j =1 , one let’s dµ be the measurewith Jacobi parameters { a j +1 , b j +1 } ∞ j =1 where we drop the first two pa-rameters and knock indices down. One can prove that the m -function,call it m of dµ is related to m by (see [674, Theorem 3.2.4]) m ( z ) = 1 − z + b − a m ( z ) (6.52) This suggests another way to recover the Jacobi parameters from dµ .From dµ , compute 1 /m ( z ) using (6.51). By (6.52), the leading Laurentseries at infinity is − z + b − a z − + O( z − ) so one finds the first twoJacobi parameters. Then use (6.52) to compute m and iterate. Theiteration of (6.52) gives the continued fraction: m ( z ) = 1 − z + b − a − z + b − a − z + b · · · (6.53) ? ? a representation that goes back to Jacobi, Markov and Stieltjes. Inparticular, Stieltjes proved that this expansion converges on the com-plement in C of the convex hull of the support of dµ ; see [680, Section7.7]. For this reason, the solution of the inverse problem (of going fromthe measure back to the Jacobi parameters from the spectral measure)is called the continued fraction approach as opposed to the first ap-proach called the OP approach.As Gel’fand-Levitan remark in their paper, their approach to theinverse problem for Schr¨odinger operators is an analog of the OP ap-proach to the inverse Jacobi matrix problem. About 1985, I beganto wonder what the Schr¨odinger operator analog was to the continuedfraction approach. [657] is one of the papers I am proudest of for thefollowing reason. I’ve felt one of my weaknesses is a lack of persistence.If I couldn’t solve some problem fairly quickly, I’d drop it and, while Imight not totally ignore it, I didn’t usually return to it without somereally good idea in advance. But this question is one I’d spend a few weeks thinking about every few years until I finally solved it 1996-97!Early on I realized that there was an analog of (6.52) for the continuumcase, namely the celebrated Ricati equation for the Weyl m -function dmdx ( z, x ) = V ( x ) − z − m ( z, x ) (6.54) ? ? where m ( z, x ) = u (cid:48) ( x, z ) u ( x, z ) (6.55) (when Im( z ) > 0) with u the solution of − u (cid:48)(cid:48) + V u − zu = 0 thatgoes to zero as x → ∞ . The issue was deciding what might be thecontinuum analog of a continued fraction and it turned out to be aLaplace transform!These papers also have results for operators on [0 , a ] with a < ∞ , butfor simplicity I will (except for some remarks) discuss the case where V is bounded, in L ([0 , ∞ )) and continuous. In that case, I proved thefollowing (cid:104) T6.13 (cid:105) Theorem 6.13 ([657]) . Let V be bounded, in L ([0 , ∞ ]) and continu-ous and let m ( z, x ) be the Weyl m -function (6.55) . Then there exists ajointly continuous function A ( α, x ) on (0 , ∞ ) × [0 , ∞ ) so that one hasthat m ( − κ , x ) = − κ − (cid:90) ∞ A ( α, x ) e − ακ dα (6.56) whenever κ > (cid:82) ∞ | V ( x ) | dx . Moreover, A ( α, x ) depends only on { V ( y ) | x ≤ y ≤ x + α } and lim α ↓ A ( α, x ) = V ( x ) (6.57) and one has that ∂A ( α, x ) ∂x = ∂A ( α, x ) ∂α + (cid:90) α A ( β, x ) A ( α − β, x ) dβ (6.58) where if V is C , then A is jointly C and this equation holds in classicalsense and in general it holds in a suitable weak sense. Remarks. 1. The solution of the inverse problem should be clear. Togo from m ( z, x = 0) to V , one uses (6.56) to obtain A ( α, x = 0) usinguniqueness of the inverse of Laplace transform ( A is determined byasymptotics of m so, one only needs (6.59) below and not the stronger(6.56)). Then using the integrodifferential equation (6.58), one finds A ( α, x ) for all x (indeed, A ( α, x ) is determined by A ( β, x = 0) for β ∈ [0 , α + x ]). Then one finds V using (6.57). The result is that A ( α, x = 0) for α < B determines V ( y ) for y < B and vice-versa. WELVE TALES 73 2. Continuity of V is not critical. One shows for discontinuous V ,one has that A ( α, x ) = V ( x + α ) + E ( α, x ) where is E is continuousand lim α ↓ E ( α, x ) = 0 so, if V is only locally L , one has that (6.57)holds in the sense of local L convergence rather than pointwise in x .Moreover, E is smoother than V , so, for example if V is locally C with kinks, A has kinks precisely along the lines where x + α is a kinkpoint.3. Simon [657] discusses V ’s that are locally L and (more or less)bounded from below. Gesztesy-Simon [225] extends the theory to ar-bitrary locally L potentials, even those which are limit circle at ∞ . Inplace of (6.56), for general V , one has the formula m ( − κ , x ) = − κ − (cid:90) a A ( α, x ) e − ακ dα + O( e − (2 a − ε ) κ ) (6.59) for all ε > , a > 0. This formula, for each x , determines A ( α, x ) from m ( α, x ). This formula alone doesn’t go from directly from A ( · , x ) to m ( · , x ) but one can do that by going through the inverse constructiondiscussed in remark 1. [225] also has an explicit way to go directly fromthe spectral measure, dρ to A ( · , x = 0), namely A ( α, x = 0) = − (cid:90) ∞−∞ λ − / sin(2 α √ λ ) dρ ( λ ) (6.60) ? ? (the integral diverges so the formula requires a proper interpretation!).Ramm-Simon [519] discuss asymptotics of A ( α, x = 0) as α → ∞ when V ( x ) has very nice behavior at infinity.One important consequence of this work is a local version of thefollowing theorem of Borg [73] and Marchenko [459]: ? (cid:104) T6.14 (cid:105) ? Theorem 6.14 (Borg-Marchenko Theorem) . Let V and V be twolocally L functions on [0 , ∞ ) and let m , m be the m -functions as-sociated to − d dx + V j . Then V ( x ) = V ( x ) for all x ∈ [0 , ∞ ) ⇐⇒ m ( z ) = m ( z ) for all z ∈ ( −∞ , with | z | large. The local version, which originally appeared in [657]: (cid:104) T6.15 (cid:105) Theorem 6.15 (Local Borg-Marchenko Theorem) . Let V and V betwo locally L functions on [0 , ∞ ) and let m , m be the m -functionsassociated to − d dx + V j . Then V ( x ) = V ( x ) for all x ∈ [0 , a ) ⇐⇒ asymptotically for z ∈ ( −∞ , one has that: ∀ ε> ∃ C,R> ∀ z ∈ ( −∞ ,R ) | m ( z ) − m ( z ) | ≤ Ce −| z | (2 a − ε ) (6.61) This follows easily from Theorem 6.13 because (6.61) is equivalentto A ( α, x = 0) = A ( α, x = 0) for all x ∈ [0 , a ] and by the differential equation, that happens if and only if V ( x ) = V ( x ) for all x ∈ [0 , a ].While this proof is quite illuminating, it does require one to develop anelaborate machinery. In [226], Gesztesy-Simon found a simple directproof of Theorem 6.15 and then Benewitz [63] showed that an argumentbased on Borg’s method in [73] provides a really short proof.One memorable aspect of this work was a talk I gave about it atRutgers while the work was being written up. I was excited to giveit because Gel’fand (of Gel’fand Levitan and other fame), who hadmoved there after he left Russia was in the audience. He asked somepointed questions, made some positive comments and, in particular,pointed out that one positive element was that it was easy to extendto matrix valued potentials. The was in the fall of 1997 when I was 51and Gel’fand was 84. I remember thinking to my self “Gee, I hope I’mthat sharp when I’m 84.” I mentally paused and then thought “No,you wish you were that sharp when you were 48!”.Using [657, 225], Gesztesy and I proved [227] that ? (cid:104) T6.15A (cid:105) ? Theorem 6.16. Let V and V be two potentials on [0 , ∞ ) so that thetwo operators − d dx + V j , j = 0 , both have discrete spectrum whichare identical. Then there is a smooth family of potentials V t ; 0 ≤ t ≤ interpolating between them which each have the same spectrum. Remarks. 1. The proof shows that if V j are both C k , then the V t ’swe construct are also C k .2. Using A ’s, the proof is almost trivial. One takes A t = tA + (1 − t ) A !3. This result is interesting because the analogous result on thewhole line is open. For example, it is not even known if the C ∞ poten-tials going to ∞ at ±∞ whose spectrum is the same as the harmonicoscillator is an connected set!Useful tools in inverse theory are so called trace formula that providethe potential as an integral of some kind of spectral or scattering object.Gesztesy and I [219] found a fairly general trace formula with furtherdevelopments in papers we wrote with others [228, 229, 230, 220]. If V is a continuous function on R bounded from below on defines H = − d dx + V and H D ; x is the same operator with a Dirichlet boundarycondition forced at x . ξ ( x, λ ) is then the Krein spectral shift (see, e.g.[680, Sections 5.7-5.8]) going from H D : x to H . ThusTr( e − tH − e − tH x ; D ) = t (cid:90) ∞ e − tλ ξ ( x, λ ) dλ (6.62) ? ? WELVE TALES 75 and the general trace formula in [219] is that V ( x ) = lim α ↓ (cid:20) E + (cid:90) ∞ E e − αλ [1 − ξ ( x, λ )] dλ (cid:21) (6.63) ? ? for an E below the bottom of the spectrum of H .Another set of inverse type problems that I studied in five paperswith Gesztesy (one also jointly with Del Rio who first told me of The-orem 6.17 below) [221, 222, 223, 137, 224] was motivated in part by aremarkable theorem of Hochstadt-Lieberman [297] (we state this withDirichlet boundary conditions and continuous V although they holdmore generally): (cid:104) T6.16 (cid:105) Theorem 6.17. Let V be a continuous on [0 , and let H be − d dx + V on L (0 , dx ) with u (0) = u (1) = 0 boundary conditions. Then the setof eigenvalues of H and V on [0 , / determine V . Typical of our results is that if one considers the H operator withboundary conditions u ( a ) = u ( b ) = 0 with 0 ≤ a < b ≤ L ( a, b ; dx ), then, for a ∈ (0 , , a ], [ a, 1] and [0 , 1] determine V [224]! The point of our papersis that eigenvalues are zeros of suitable m -functions and factorizationtheorems for analytic functions together with known asymptotics (andsome partial information on V) determine m and so by a version ofBorg-Marchenko for bounded intervals they determine V . For someresults, we also use Phragmen-Lindelof theorems to allow us to onlyneed information on some of the eigenvalues.I close this subsection on scattering theory ideas by mentioning twoof my papers that have somewhat unusual applications of the traceclass completeness theory (the Kato-Rosenblum theorem mentionedearlier in this section). One of these exploits an extension of the Kato-Rosenblum trace class scattering theorem due to Pearson [502]: ? (cid:104) T6.17 (cid:105) ? Theorem 6.18. Let A and B be two self-adjoint operators and J abounded operator so that AJ − J B is trace class. Then Ω ± ( A, B : J ) = lim t →∓∞ e itA J e − itB P ac ( B (6.64) exist. Remarks. 1. For a proof and a discussion of implications, see Reed-Simon [527, Theorem XI.7].2. If A and B are unbounded, one has to worry about the meaningof “ AJ − J B is trace class”. The more precise hypothesis is that there is a trace class operator C so that for all ϕ ∈ D ( A ) and ψ ∈ D ( B ), onehas that (cid:104) ϕ, Cψ (cid:105) = (cid:104) Aϕ, J ψ (cid:105) − (cid:104) ϕ, J Bψ (cid:105) 3. In applications, it is a useful (and easy to prove) fact that if J iscompact, the limit in (6.64) exists and is 0.4. One example that shows the power of Pearson’s extension is that itimplies that if ( A + i ) − − ( B + i ) − is trace class, then the ordinary waveoperators, (6.39), exist and are complete (a result sometimes calledthe Birman-Kuroda theorem). For this assumption implies the one ofPearson’s theorem with J = ( A + i ) − ( B + i ) − . Then using remark3, we get existence for J = ( B + i ) − . Applying that limit to vectorsof the form ( B + i ) ψ proves existence of the ordinary wave operatorsand then Theorem 6.10 implies completeness.5. The following extension (and corollary of) of Pearson’s theoremis useful: if ( A + i ) − [ AJ − J B ]( B + i ) − (6.65) ? ? is trace class, then Pearson’s theorem applies with J = ( A + i ) − J ( B + i ) − and, by mimicking the argument in remark 4, one sees that thelimits in (6.64) exist.Davies and I [126] studied very general 1D Schr¨odinger operators, H = − d dx + V . Let J r be a smooth function which is 0 on ( −∞ , − , ∞ ) and J (cid:96) = 1 − J r . Then under great generality on V (e.g.any bounded V ; see below), one sees that Pearson’s theorem impliesthat Ω ± ( H, H : J r ) and Ω ± ( H, H ; J (cid:96) ) = 1 − Ω ± ( H, H : J r ) exist and itis not difficult to show that they are projections P ± r and P ± (cid:96) . Letting H ± r = ran P ± r (and similarly for (cid:96) ), one sees that: ? (cid:104) T6.18 (cid:105) ? Theorem 6.19 (Davies-Simon [126]) . Let V be a potential on R whosepositive part is in L loc and negative part is a form bounded perturba-tion of − d dx with relative bound less than . Let H ac be the absolutelycontinuous subspace for H . Then H ac = H − r ⊕ H − (cid:96) = H + r ⊕ H + (cid:96) (6.66) ? ? Moreover if ϕ ∈ H − r , then as t → ∞ , one has that for any R theprobability that e − itH ϕ lies in { x | x < R } goes to zero and similarly forthe other four subspaces. Remarks. 1. For discussion of form bounded perturbations, see, forexample, [680, Section 7.5]. Under the hypotheses, one can define H as a closed form sum on Q ( − d dx ) ∩ Q ( V + ). WELVE TALES 77 2. What this theorem says is that every ϕ ∈ H ac is the sum of apiece that goes in x to plus infinity as t → ∞ and a piece that goes tominus infinity. Similarly, there is a decomposition as t → −∞ .3. There are also results in [126] on operators on R ν which areperiodic in all directions but one.4. [126] also has a powerful method for sometimes eliminating s.c.spectrum called the twisting trick. I will not say more about it exceptto note that subsequently, Davies [124] used a variant for a penetratinganalysis of double well Hamiltonians, a subject I will discuss further inSection 8.Once you have this set up there is a natural notion of reflectionless:we say that a potential is reflectionless if and only if H + (cid:96) = H − r , thatis all states that come in from the left go out entirely on the rightwith no reflection. Deift and I [135] conjectured in 1983 that a 1Dalmost periodic Schr¨odinger operator is reflectionless in this sense. Adifferent notion of reflectionless arose in the theory of solitons (see, forexample, [489, Chapter II]) and there has arisen a huge literature onthis notion capped by Remling’s characterization [534] of right limitsof a.c. spectrum mentioned in the remarks after Theorem 6.3. Namely,a 1D Schr¨odinger operator, H , is called spectrally reflectionless if andonly if for all x and Lebesgue a.e. E in the a.c. spectrum of H , one hasthat (with G the Green’s function, i.e. integral kernel of the resolventof H ) lim ε ↓ Re G ( x, x ; E + iε ) = 0 (6.67) ? ? In 2010, more than 25 years after the conjecture of Deift-Simon,Breuer, Ryckman and Simon [77] proved that conjecture by provingthe much more general (they also have this result for Jacobi matricesand two sided CMV matrices): ? (cid:104) T6.19 (cid:105) ? Theorem 6.20 ([77]) . A one dimensional Schr¨odinger operator is spec-trally reflectionless if and only if it is reflectionless in the sense ofDavies-Simon. The other trace class paper that I should mention is by Simon andSpencer [686]. Typical of our results is (cid:104) T6.21 (cid:105) Theorem 6.21 ([686]) . Let h be a discrete Schr¨odinger operator on R so that lim sup n →∞ | b n | = lim sup n →−∞ | b n | = ∞ . Then h has no a.c.spectrum. Remarks. 1. I want to emphasize this involves lim sup rather than lim.If it were lim, the spectrum would be discrete but it is easy to constructexamples with the lim sup condition but spectrum all of R . This result says that tunnelling through high barriers destroys ac spectrum whichis an intuitive result.2. The proof is an easy application of the Birman-Kuroda theorem.One picks a two sided subsequence, { n j } ∞ j = −∞ going to ±∞ as j → ±∞ and so that (cid:80) ∞ j = −∞ | b n j | − < ∞ and by simple estimates shows that if h ∞ is h with the sites at all n j removed (what you get if b n j were takento ±∞ ), then ( h + i ) − − ( h ∞ + i ) − is trace class. So since h ∞ , as adirect sum of finite matrices, has no ac spectrum, neither does h .3. Simon-Spencer [686] use this idea in many other ways. Not onlythe obvious ones like continuum Schr¨odinger operators where the bar-riers not only have to be high but also not too narrow but also somelimited results in higher dimensions. There is even a proof that certainone dimensional random discrete Schr¨odinger have no ac spectrum (aswe’ll recall in Section 11, more is true).4. Once I started working in the theory of orthogonal polynomials, Ilearned that Dombrowski [158] had the same idea eleven years prior tous in a different but closely related (and also simpler!) context. Namely,she considered Jacobi matrices, J , and proved that if lim inf n →∞ a n = 0,then J has no ac spectrum. For one picks a subsequence n j so that (cid:80) ∞ j =1 a n j < ∞ . Dropping those a n j ’s gives a trace class perturbationwhich turns the matrix into a direct sum of finite matrices and one usesthe Kato-Birman theorem!I close this section on N-Body Quantum Mechanics by mentioningthat my books by Reed-Simon [526, 527, 528] and Cycon et al. [117]have been a useful introduction to many researchers trying to learnabout the subject.7. Magnetic Fields in NRQM (cid:104) s7 (cid:105) One of the things that made the 70’s so fruitful for my researchis that I kept finding subareas where mathematical physicists hadn’tlooked, so I could wander around an orchard and pick the low hangingfruit and write papers which, because they set the framework, couldbe widely used and quoted later. One of these areas was physics ofNRQM with magnetic fields where Avron Herbst and I wrote a seriesof papers [30, 31, 32, 33] with lots of intriguing results, a major partof this section. The basic objects studied are magnetic Hamiltonians H ( a , V ) = − ( ∇ − i a ( x )) + V ( x ) (7.1) ? ? It may seem strange to imply that there wasn’t earlier work on thissubarea given that, for example, fifteen years before, Ikebe-Kato [320] WELVE TALES 79 had proven a fairly general result on self-adjointness in magnetic field,at least when the vector potential, a j ( x ), is C . Kato, as discussed inthe last Section, included magnetic fields in his Kato inequality paperas did some of my followup work. But this earlier work focused almostentirely on self-adjointness or issues where one treated the magneticvector potential as just a coefficient in the PDE but didn’t focus onthe physics underlying the magnetic field nor the special roles of gaugeinvariance nor the role of the non-commutativity of the componentsof − i ∇ j − a j . We were really the first researchers to look at theseproblems as mathematical physicists rather than as mathematicians.Avron and Herbst were following up on their earlier beautiful workon constant electric field [27]. They may have asked me to join them formagnetic fields because of my earlier work about a year before relatedto Theorem 6.9. To begin with, I proved (cid:104) T7.1 (cid:105) Theorem 7.1 ([601]) . Let E ( a , V ) = inf ϕ ∈ L ( R ν ) (cid:104) ϕ, H ( a , V ) ϕ (cid:105) (7.2) ? ? Then for any a , V , we have that E ( a , V ) ≥ E ( a = 0 , V ) (7.3) Remarks. 1. “any a , V ” of course means for which one can reasonablydefine the operators and for which they are bounded below.2. In other words, energies go up if one turns on any magnetic field.3. Formally, the proof is easy: if ϕ = | ϕ | e iψ , then ( ∇ − i a ) ϕ =( ∇ | ϕ | − i a | ϕ | + i ∇ ψ | ϕ | ) e iψ so | ∇ − i a ) ϕ | ≥ ∇ | ϕ | . Squaring andintegrating, one gets (cid:104) ϕ, H ( a , V )) ϕ (cid:105) ≥ (cid:104)| ϕ | , H ( a = 0 , V )) | ϕ |(cid:105) (7.4) from which (7.3) results. A complete proof isn’t much harder.Since (7.4) involves | ϕ | and if ϕ is antisymmetric in particle co-ordinates, | ϕ | is symmetric, this argument works for bosons but notfermions. Moreover, if a particle has spin, one adds a σ · B term whichdestroys (7.3), and the theorem only holds for spinless particles (in fact,Lieb (his result and proof were published as an appendix to [30]) provedin constant magnetic field for the Pauli equation of spin electrons,energies went down when magnetic fields are turned on; for a whilethere was a conjecture that for the Pauli equation Theorem 7.2 belowheld with the direction of the inequality reversed for general magneticfield but Avron and I [40] found a counterexample) so I wrote a paperentitled “Universal Diamagnetism for Spinless Bosons” and submittedit to Phys. Rev. Lett. The referees report was memorable. Essentially it said Since there are no stable spinless bosons in nature, the resultof this paper is of limited physical applicability. But it is nice to seesomething nontrivial proven in just a few lines, so this paper should beaccepted as an example to others. So the paper was accepted [601]!The next step illustrates the dynamics of the brown bag lunch atPrinceton. At one, I described Theorem 7.1 and mentioned that I con-jectured that this was a zero temperature result and that there shouldbe a finite temperature result that was an inequality between integralkernels of semigroups, and I was working on it. Almost immediately, EdNelson interjected: “You know that follows from the stochastic integralmagnetic field version of the Feynman Kac formula.” Stirred by this,I found a direct proof from Kato’s inequality which, in typical fashion,Ed refused to be a coauthor of. These inequalities which I dubbed dia-magnetic inequalities are used often in the study of quantum mechanicsin magnetic fields (some tried to call them Nelson-Simon inequalitiesbut my name was catchier). They say: (cid:104) T7.2 (cid:105) Theorem 7.2 (Diamagnetic Inequalities) . Let a ∈ L loc , V + ∈ K locν , V − ∈ K ν . Then C ∞ ( R ν ) is a form core for H ( a , V ) and pointwise | exp[ − tH ( a , V )] ϕ | < exp[ − tH ( a = 0 , V )] | ϕ | (7.5) ? ? The reader will recognize that this follows from Theorem 6.9 and(6.32). Of course, this is only if a is smooth. The full result I provedin Simon [614] and is obtained by using the Trotter product formulaand gauge transformations. Many years later, Hundertmark and I [313]wrote a paper that proved diamagnetic inequalities for differences ofsemigroups with Neumann and Dirichlet boundary conditions. Thisallowed a quick proof of invariance under change of boundary conditionof the density of states in magnetic field, something whose prior proofshad been complicated. This paper also had a new quadratic formsproof of diamagnetic inequalities.I turn now to the results in the papers with Avron and Herbst. Afraction of them specifically discuss constant magnetic field. Classi-cally, 2D electrons in such a field go in circles in two dimensions whileis 3D they can move out to infinity in the direction of the field. Quan-tum mechanically, one has the celebrated Landau levels - in 2D purepoint spectrum (although of infinite multiplicity). As we’ll see this pro-duces various enhanced binding scenarios. Here are some of the mainresults.1. Center of Mass Reduction . The physics of N particles in constantmagnetic field is invariant under translations of all of the particle coor-dinates but the mathematics of the reduction can be very different from WELVE TALES 81 what is described in Theorem 6.1. If the total charge is not 0, the uni-taries associated with translations in different directions perpendicularto the field no longer commute but only commute up to a phase. Putdifferently, the components of the conserved generators of translation(called quasimomentum) obey canonical commutation relations. It issurprising that this subject wasn’t worked out in the physics literaturebefore us, but it wasn’t. Many of the citations of this paper are inthe physics literature as seen by the fact that it has 355 citations onGoogle Scholar but only 26 on MathSciNet!2. Borel summability and dilation analyticity of atoms in constantmagnetic field . In paper III [32], Borel summability of the perturba-tion series in magnetic field strength for constant field is proven foreigenvalues of multielectron atoms which are discrete and simple onthe space where L z , the total azimuthal angular momentum about theaxis of the field, is fixed. Paper I [30] discusses dilation analyticityfor hydrogen in a constant magnetic field and paper III [32] discussesthis for multielectron atoms. Stability is a key technical issues. Theseresults (and more for hydrogen in constant field) were announced in[28].3. Large B . Paper IV [33] discusses asymptotic behavior for hydrogenin a large constant field B as | B | → ∞ . Because terrestrial magneticfields in natural units are tiny, this is of interest only in astrophysicalcontexts. If H ( B ) = ( − i ( ∇ ) − B ˆ z × r ) (7.6) H ( B, λ ) = H ( B ) − λ | r | − (7.7) ? ? then, by scaling H ( B, z ) is unitarily equivalent to B ( H (1 , zB − / )), solarge B with z = 1 is equivalent to studying fixed field and small λ .Since the magnetic field binds in the two dimensions perpendicular to B , this is effectively a 1D weak coupling problem where I’d discoveredsmall coupling asymptotic series (see the discussion in Section 8) butthe decay is slower than in my earlier work. However, one can modifythat work and obtain the first few terms which are very complicated(the leading order is log( B ) as first suggested by Ruderman [546]) butthere are log(log( B )) and log( B ) − terms!4. The Hydrogen Zeeman Ground State has L z = 0. In paper III[32], we proved that for hydrogen in magnetic field, the ground statehas L z = 0. For zero magnetic field, the ground state is positive (see,for example, [527, Section XIII.12]) so in an azimuthally symmetricpotential, the ground state has L z = 0. But paper I [30] constructsexamples of azimuthally symmetric potentials so that for small field the lowest part of the spectrum is discrete and the lowest energy isnot L z = 0 (Lavine-O’Carroll [422] also have such examples). Thedeep result that for attractive Coulomb potentials the ground state has L z = 0 uses the monotonicity of V ( r ) in r and correlation inequalitiesas extended to quantum systems as discussed in Section 3.5. Enhanced Binding . As discussed in Section 8, if V ≤ C ∞ ( R ), then for small λ , the operator − ∆ + λV has no eigenvaluesin ( −∞ , 0) but for C ∞ ( R ), − d dx + λV always has at least one negativeeigenvalue. Things are different if − ∆ is replaced by H ( B ) (given by(7.6)) with B (cid:54) = 0. For H ( B ) = (cid:101) H ( B ) ⊗ + ⊗ (cid:16) − d dx (cid:17) (7.8) ? ? on L ( R ) = L ( R ) ⊗ L ( R ) where (cid:101) H ( B ) has point spectrum { (2 n + 1) B } ∞ n =0 , each of infinite multiplicity.Further analysis (see [30, Section 3]) shows that if one restrictsto L z = m , one has point spectrum { − m, 0) + n + 1 } ∞ n =0 ≡{ E m ( n ) } ∞ n =0 , each of multiplicity 1. Thus H ( B ) (cid:22) L z = m is a directsum of copies of − d dx + E m ( n ). If V ≤ L z ) each space with L z = m and m ≥ H ( B ) + V below its essential spectrum, so we conclude that ? (cid:104) T7.3 (cid:105) ? Theorem 7.3 (Avron, Herbst and Simon[30]) . If V ∈ C ∞ ( R ) is anon-negative (not identically ) function of only z and ρ = (cid:112) x + y ,then H ( B ) + V for any B (cid:54) = 0 has essentially spectrum [ | B | , ∞ ) andinfinitely many eigenvalues in ( −∞ , | B | ) . Remark. By diamagnetic inequalities, if λ is so small that − ∆ + λV has no negative spectrum than all the eigenvalues are in [0 , | B | ) forall B (cid:54) = 0. Moreover, for B fixed, for all small λ there is exactly oneeigenvalue for each of H ( B ) + V (cid:22) ( L z = m ) for each m ≥ B > Negative Ions . One form of enhanced binding discovered byAvron-Herbst-Simon involves negative ions in magnetic field. In na-ture, most neutral atoms do not bind any extra electrons althoughthere is no rigorous proof that this is even true for He − . By usingideas discussed in point 5 above, AHS in an announcement [29] inPhys. Rev. Lett. and in paper III [32] prove that every neutral atomicHamiltonian in non-zero constant magnetic field will bond at least oneadditional electron.7. Magnetic Bottles . In paper I [30], we looked at the question ofwhether one can produce Hamiltonians with compact resolvent withjust magnetic field alone (i.e. [ H ( a, V = 0) + 1] − compact). It is easyto see how to do this in even dimension, e.g. R by making B z → ∞ as WELVE TALES 83 x + y → ∞ . (even though there is no z direction, B z ≡ ∂ x a y − ∂ y a x is defined.) , but apriori, it is not clear how do this in odd dimension.However, we found lots of examples, e.g. in 3D, B = ( x, y, − z ) (since ∇ · B = 0, there is a with B = ∇ × a ).8. General Spectral and Scattering Theory in constant magnetic field .In papers I, II and III [30, 31, 32], Avron, Herbst and I tried to extendmuch of the theory of 2 − and N − body systems when − ∆ is replacedby H ( B ) (given by (7.6)). Much of it is straightforward but thereare interesting twists. For example, for perturbations, V of − ∆, theAgmon-Kato-Kuroda theory needs (1+ | x | ) − − ε decay but the (2-body)analog for H ( B ) developed in [30] only needs (1 + | z | ) − − ε decay andany kind of decay in the orthogonal directions. Cook’s method asdeveloped in [30] for existence of Ω ± ( H ( B ) + V, H ( B )) needs (1 + | z | ) − − ε decay in V but allows growth (!) in the x and y directions byless than an inverse Gaussian since Landau levels decay in a Gaussianmanner! Because of the unusual form of reduction of the center of massdiscussed in [31], the HVZ theorem proved there is more involved thanin the zero field case.I turn now to some later work on NRQM in magnetic field. AHS,because of its focus on the constant field case, left an important issue onthe table. The standard analysis using Weyl’s criterion [680, Problem3.14.5] and localized test functions proves that σ ess ( H ( a , V )) = [0 , ∞ )if a ( x ) → V ( x ) → x → ∞ . But this gets the physics wrong.If say ν = 2 and B z ( x, y ) = C (1 + ρ ) − α (7.9) then in any fixed gauge, a → α > σ ess ( H ( a , V )) = [0 , ∞ ) so long as α > 0, i.e. rather than requiring a → 0, we should only need B → ∞ . The first theorem of thistype was proven in the PhD. thesis of my student Keith Miller [470]which he never published since he decided not to take an academicjob. He used test functions and Weyl’s criterion but functions with an x -dependent phase factor that implements a change of gauge in whichthe new a is small on the support of the test function. The standardreference for this is a joint paper that he and I wrote [471] which Iwill turn to shortly. There is now a huge literature on the this issuewhich is summarized in Last-Simon [416], a paper that has an HVZtype result in terms of limits at infinity of magnetic fields (we limitedourselves to bounded, uniformly H¨older continuous magnetic fields).Miller and I [471] found the following remarkable fact: ? (cid:104) T7.4 (cid:105) ? Theorem 7.4 (Miller-Simon [471]) . Let H ( α ) be the quantum Hamil-tonian of a D particle with V = 0 and magnetic field given by (7.9) .For all α > , σ ( H ( α )) = [0 , ∞ ) . If < α < , H ( α ) has dense purepoint spectrum in all of [0 , ∞ ) . If α = 1 , there is E (depending on C )so that the spectrum is dense pure point on [0 , E ] and purely a.c. on [ E , ∞ ) . If α > , the spectrum is purely a.c. Remark. There is an arithmetic mistake in the calculation of E in[471] that was recently noted and corrected by Avramska-Lukarska etal [25].The proof is easy. That σ ( H ( α )) = [0 , ∞ ) follows from Miller’sargument about B → 0. On the other hand since B is azimuthalsymmetric, one can pick a gauge in which H ( a , V ) commutes withrotations and look at fixed L z = m where the operator is unitarilyequivalent to − d dx + V α,m ( x ) on L ([0 , ∞ )). If 0 < α < 1, then V α,m →∞ so each H ( α ) (cid:22) L z = m has purely discrete spectrum (althoughthey have to fit together to give dense point spectrum)! If α = 1, each V α,m → E and if α > 1, each V α,m → α > 1, theclassical orbits are all unbounded, if 0 < α < 1, all orbits are boundedwhile if α = 1 the orbits are either bounded or unbounded dependingon whether E < E or E > E .While the physics is not related, there is an intriguing result ofHempel et. al [286] that has similar mathematics. H = − ∆ + cos | x | on L ( R ) has alternating bands of a.c. and dense point spectrum!The argument is similar to that of [471]: near x = ( x, , 0) with x large, H looks like − d dx + cos x − d dy − d dx which lets one show that σ ( H ) = [ E , ∞ ) for suitable E . On the other hand, if S is the (band)spectrum of − d dx +cos x on L ( R ), the restriction of H to each fixed an-gular momentum space is − d dr + (cid:96) ( (cid:96) +1) r + cos r which has a.c. spectrumon S and eigenvalues in the gaps.Finally, I have two papers [47, 629] that looked at continu-ity in B of various objects associated to H ( B ) + V when V is periodic. My interest was originally sparked by a lack ofcontinuity in frequency, α , of the density of states (and thespectrum) of the almost periodic Jacobi matrix (see section 9), Hu ( n ) = u ( n + 1) + u ( n − 1) + λ cos( παn + θ ) u ( n ) which is supposedto be a strong coupling approximation of a 2 D periodic operator inmagnetic field (Thouless explained the apparent puzzle of the continu-ity of the density of states shown in [629] and this lack of continuity WELVE TALES 85 of the above H in α for fixed θ . The correct analog is not the densityof states associated to the operator H for fixed θ . Rather the correctanalog is the integral over θ and this is known to be continuous in α [46,Theorem 3.3]). The first paper [629] proves continuity of the density ofstates in B . The key is to note that while a in any fixed gauge is mis-behaved at infinity the diagonal heat kernel e − t ( H ( B )+ V ) ( x, x ) is gaugeinvariant and so periodic in x . This yields continuity of the Laplacetransform of the density of states. [47] deals with the more subtle issueof continuity of the spectrum in B .8. Quasi-classical and Non–quasi-classical limits (cid:104) s8 (cid:105) The structures of quantum and classical mechanics are very differ-ent, so it is remarkable that a world we believe is described by quantumtheory is consistent in the right realm with classical mechanics. Thustheir connection has been a compelling subject for both physicists andmathematicians. I have a lot of work of work that explores their con-nection and when the naive connection needs modification.One of the simplest connections goes back to Weyl [732] in 1912, anumber of years before the new quantum mechanics and so, obviously,done in different context, namely classical electromagnetism. Lookingat − (cid:126) ∆ below a fixed energy E at small (cid:126) is the same as looking at − ∆ below a very large energy. Weyl fixed a compact set Ω ⊂ R ν withsmooth boundary and looked at the number, N Ω ( E ), of eigenvalues be-low E for − ∆ Ω D , the Laplacian in Ω with Dirichlet boundary conditionsand proved that lim E →∞ N Ω ( E ) E ν/ = τ ν (2 π ) ν | Ω | (8.1) where τ ν is the volume of the unit ball in R ν and | Ω | is the volume ofΩ (an exposition of the proof of (8.1), close to Weyl’s, can be found,for example, in [680, Section 7.5] which also explains the interestinghistory of how Weyl came to consider this problem). The right sideof (8.1) has two volumes in R ν and can be interpreted as a volumein phase space. − ∆ Ω D is the quantum Hamiltonian when the units are (cid:126) = 1 = 2 m , so E = p and the volume in phase space of { ( x, p ) | x ∈ Ω , p ≤ E } is τ ν E ν/ . Thus (8.1) says that for large E , N Ω ( E ) lookslike the volume in phase space where the energy is less than E times(2 π ) − ν . (cid:126) = 1 ⇒ h = 2 π so this says each state takes a volume of h ν .We now shift to a particle with interaction. As above, − (cid:126) ∆ + V inthe small limit is related to − ∆ + λV in the large λ limit. If we areinterested in the number of negative energy states, the relevant volume is { ( x, p ) | p + λV ( x ) ≤ } , so the semiclassical number of states is N cl,V ( λ ) = τ ν λ ν/ (2 π ) ν (cid:90) R ν | V ( x ) | ν/ d ν x (8.2) ? ? From early on, while a graduate student, I had an interest in boundson the number of bound states of quantum systems, although I didn’tinitially think about quasi-classical limits. I found in the literature twowell-known results. Bargmann [53] proved that for − d dx + V ( x ) on L (0 , ∞ ) with u (0) = 0 boundary conditions, the number of negativeeigenvalues, n ( V ), obeys n ( V ) ≤ (cid:90) ∞ r | V ( r ) | dr (8.3) (Bargmann, who viewed this as a bound are s -waves for a 3 D problemwith a central potential, also had results for higher angular momen-tum). Schwinger [557] (see below for the work of Birman) proved on L ( R ), that one has that N ( V ), the number of negative eigenvalues(counting multiplicity) of − ∆ + V , is bounded by N ( V ) ≤ π ) (cid:90) | V ( x ) || V ( y || x − y | d xd y (8.4) With this in mind, as a graduate student, I realized that the growthof N ( λV ) as λ → ∞ was interesting and not what one might expectnaively from (8.4) when ν = 3. Rather I found [580] when V is verynice there are λ ν/ upper and lower bounded (but I did not prove strict λ ν/ asymptotic behavior nor did I realize at the time the connectionto quasi-classical behavior).Quoting my paper, several years later, Martin [460] proved the muchstronger quasi-classical resultlim λ →∞ N ( λV ) N cl,V ( λ ) = 1 (8.5) on R ν when V is H¨older continuous (as noted earlier, Birman-Borzov[70], Robinson [538] and Tamura [709] proved the same result (somewith somewhat weaker hypotheses on V ) in a similar time frame).These authors all used a variant of Weyl’s argument. Interestinglyenough, the result without proof (essentially doing a quasi-classicalcomputation and asserting its correctness) appeared in 1948 as a solvedproblem in the quantum mechanics book of Landau-Lifshitz [410, Sec-tion 48, Problem 1] (I could only check this in the second edition ofthe English translation; 1948 is the date of the first edition).I pause in the discussion of (8.5) for a side trip to the tool behindthe next steps. In 1976, Valya Bargmann reached the age of 68 and WELVE TALES 87 had to retire and the remaining senior joint appointments in math-ematical physics at Princeton edited a festschrift in his honor [445].I wrote two reviews for that book on subjects where Bargmann hadbeen a pioneer, one [606] on how to go from time automorphisms toHamiltonians in Quantum Mechanics (where the foundational work wasdone by Bargmann and Wigner) and one on bounds on the number ofbound states [605]. While the second was there because of Bargmann’sbound, an especial point concerned the bound then universally associ-ated in the West with Schwinger. I had found that in the same yearas Schwinger’s paper, Birman had published [68] a long paper thatincluded the same bound as Schwinger proven by the same method.This method considers eigenvalues E < H + V where σ ( H ) ⊂ [0 , ∞ ) and V is relatively form compact and self-adjoint so, if V / ≡| V | / sgn( V ), then K E = −| V | / ( H − E ) − V / (8.6) ? ? is compact. One shows that the dimension of the solutions of K E ϕ = ϕ is the multiplicity of E as an eigenvalue of H + V (essentially becauseif ϕ = | V | / ψ , then K E ϕ = ϕ ⇐⇒ ( H + V ) ψ = Eψ ).Birman and Schwinger had the further idea of adding a couplingconstant, λ , and looking at eigenvalues, E j ( λ ), of H + λV . Using thefact that E j ( λ ) is a strictly monotone function of λ , one proves that (cid:104) T8.1 (cid:105) Theorem 8.1. The number of eigenvalues of H + V less than E < (counting multiplicity) is the number of eigenvalues (counting multi-plicity), µ > of K E . Remarks. 1. Birman and Schwinger only considered the case where V ≤ K E is self-adjoint and the compact operator, K E is self-adjoint so its eigenvalues are real. But one can prove in general thatthe eigenvalues are real even though K E may not be self-adjoint.2. µ = λ − .3. By taking limits, one can show that if K E has a limit, K , as − E ↓ 0, then the number of eigenvalues E < µ > K . When V ≤ K E ≥ 0, this in turn isbounded by Tr( K ) and Tr( K ). This provides a new proof of (8.3)and the first proof of (8.4).I felt it important to get Birman some credit for what was knownas Schwinger’s bound so in [605], I dubbed Theorem 8.1 the Birman-Schwinger principle and thereafter K E became known as the Birman-Schwinger operator (or when written as an integral operator, Birman-Schwinger kernel ) and 8.4 became the Birman-Schwinger bound . I am very glad I succeeded in this. There is an interesting postscript:several years afterwards I got a letter from Birman thanking me severaltimes for mentioning his work but then essentially asking “but why didyou include Schwinger – my paper was dated almost a year earlier”.While Birman was correct about submission dates, the result alreadywidely had Schwinger’s name and there were rumors this was one ofmany things that Schwinger had written in the notebooks he kept whileworking on radar during the Second World War and doing real physicsin his spare time.I return to my analysis of (8.5). All prior results that I knew ofrequired V to at least be continuous so I wondered about V ’s withsingularities and, more generally only L p conditions. In [604], I provedseveral theorems and conjectures about this situation. In particular, Ishowed that (cid:104) T8.2 (cid:105) Theorem 8.2 ([604]) . Let B be a Banach space of functions on R ν inwhich C ∞ ( R ν ) is dense and with (cid:107)·(cid:107) ν/ ≤ C (cid:107)·(cid:107) B for some C . Supposeone has a bound of the form N ( V ) ≤ c (cid:107) V (cid:107) ν/ B (8.7) Then (8.5) holds for all V ∈ B . The proof is by an approximation argument using the fact that if A and B are self-adjoint operators and N ( · ) the number of negativeeigenvalues (counting multiplicity), then N ( A + B ) ≤ N ( A ) + N ( B ). Ialso noted that because of Theorem 8.1, by looking at λV as λ → ∞ ,a bound like (8.7) is equivalent to µ n (( − ∆) − / | V | / ) ≤ c n − /ν (cid:107) V (cid:107) / B (8.8) I developed a version of weak trace ideals analogous to weak L p spaces(related ideas were already in Goh’berg-Krein [243]). Thinking of( − ∆) − / as a function of the Fourier transform variable, (8.8) is im-plied by (called Conjecture 2 below) (cid:107) f ( p ) g ( x ) (cid:107) ν,w ≤ c (cid:107) f (cid:107) ν,w (cid:107) g (cid:107) ν (8.9) where (cid:107)·(cid:107) p,w on the left side is a weak trace ideal norm and on the righta weak L p norm ([679, Section 2.2]). In [604], I conjectured (8.9) for2 < p < ∞ and noted that it implied for ν ≥ ⇒ Conjecture 1, the latter was of interesteven if proven by other means) N ( V ) ≤ c ν (cid:107) V (cid:107) ν/ ν/ (8.10) WELVE TALES 89 which I separately conjectured. And I note that by Theorem 8.2, thisimplies (8.5) for the maximal set where V ∈ L ν/ ( R ν ).I was already familiar with bounds a little like (8.9). In [561], ErhardSeiler and I had proven that for 2 ≤ q < ∞ one has that (cid:107) f ( p ) g ( x ) (cid:107) q ≤ C (cid:107) f (cid:107) q (cid:107) g (cid:107) q (8.11) ? ? By using interpolation ideas and this bound, I succeeded in [604] inproving (8.7) for (cid:107)·(cid:107) B = (cid:107)·(cid:107) ν/ ε + (cid:107)·(cid:107) ν/ − ε for all small ε > ε ↓ 0) and so (8.5) for V ∈ L ν/ ε ∩ L ν/ − ε but I couldn’t prove (8.9).I did this work in the spring of 1975. In the fall, Charlie Fefferman,who I’d told about my conjecture 2, introduced me to Michael Cwikel, avisitor at IAS, whom he described as an expert on interpolation theoryand might be the one to solve my conjecture. I took Cwikel aside anddescribed the problem to him.A couple of months later, I was leaving my physics office planningto check my math mailbox on my way home. As I passed his office,Elliott Lieb beckoned to me saying something like “You know yourconjecture. I think I’ve solved it.”, He proceeded to describe to mehis beautiful proof of Conjecture 1 [434, 435] using path integrals. Ithen went to my math mailbox and found a note from Cwikel sayinghe proven Conjecture 2 and thereby Conjecture 1 enclosing a sketchof his proof [115]. I couldn’t imagine my finding Lieb’s tour de forcebut found it ironic that I failed to find Cwikel’s proof because I onlythought of using interpolation theory while Cwikel, who was an experton interpolation was smart enough to instead use in a clever way astandard harmonic analysis trick of breaking a function into the setswhere it lies between 2 k and 2 k +1 that I’d seen Stein use many timesin grad courses I’d taken not long before. I should mention here twopapers with improved versions of Cwikel’s estimates: Frank [189] andHundertmark et al. [310].In July of 1976, I went to a conference in the Soviet Union (one ofonly two trips I made there) which was ideal in terms of location andmy interest. The conference was in a small town outside Leningradbut organized by the Moscow based group of Dobrushin and Sinai soI could talk to them about the work on phase transitions describedin Section 5. And because it was near Leningrad, Birman and hisgroup could come out to meet me (I only learned later, it was not easyfor them to get permission to do so). They began by saying that mypaper [604] was very interesting but, while they didn’t quite have acounterexample, they were fairly sure that my conjecture was wrong atwhich point I told them about Cwikel and Lieb. There was confusion because the conjecture they meant was Conjecture 2 but I thoughtthey meant Conjecture 1! In fact, they handed me a reprint in Russianof a paper of Rozenblum [543] who had announced a result equivalentto my Conjecture 1 in 1972 (a detailed exposition only appeared aftermy visit [544]). Eventually, I gave the bound the name CLR bound ,a name which stuck. I believe that Rozenblum feels this is unfair butgiven the methods are totally different and the work independent, Ithink it appropriate. Before leaving this subject, I should mention alater different proof of Li-Yau [432] and a non-path integral variant ofLieb’s proof by Rozenblum-Solomyak [545].In the same time frame as my conjecture, Lieb and Thirring [446], aspart of their brilliant proof of the stability of matter, exploited anotherquasi-classical bound, namely, if E j ( V ) are the negative eigenvalues of − ∆ + V on L ( R ), then (cid:88) j ( − E j ( V )) ≤ c , (cid:90) | V ( x ) | / d x (8.12) Interestingly, their proof only relied on the Birman-Schwinger bound(8.4) even though (8.12) has the right large coupling behavior and (8.4)does not.In the same Bargmann festschrift referenced above, they [447] ex-ploited the same proof to show what are now called Lieb-Thirringbounds (cid:88) j ( − E j ( V )) p ≤ c p,ν (cid:90) | V ( x ) | p + ν/ d x (8.13) for − ∆ + V on L ( R ν ) so long as p > ν ≥ p > / ν = 1 (8.14) ? ? The CLR bounds are not included but are at the borderline. As we’llsee below, there cannot be a bound at ν = 2 , p = 0 nor for ν = 1 , p < / 2. That left the case ν = 1 , p = 1 / λ →∞ λ − ( p + ν/ (cid:80) j ( − E j ( λV )) p is a universal, computablenumber c c(cid:96)p,ν , so clearly, c c(cid:96)p,ν ≤ c p,ν . It is also not hard to see that thebound on − E implied by (8.13) implies a Sobolev inequality, so the WELVE TALES 91 known best constants in the Sobolev inequalities implies another lowerbound, c Sobp,ν . Lieb-Thirring [447] conjectured that c p,ν = max( c c(cid:96)p,ν , c Sobp,ν )and using KdV sum rules (see (13.1)) proved this for p = 3 / , ν = 1(where it happens that c c(cid:96)p,ν = c Sobp,ν ). Later, Aizenman-Lieb [9] provedthat if c p ,ν = c c(cid:96)p ,ν for some p than the same is true for all p ≥ p .In particular, the Lieb-Thirring conjecture holds when ν = 1 for p ≥ p = 0 , ν ≥ ν ≥ p . That leaves ν = 1, where < p < is open (andwhere c Sobp,ν > c c(cid:96)p,ν .) This interested me so much that it is an entry inmy 2000 open problems list [658, Problem 15] of 2000. See Frank et.al. [191] for more on Lieb-Thirring inequalities.Around 2000 and for several years afterwards, I returned to issuesconnected with the critical 1 D (i.e. ν = 1 , p = ) Lieb-Thirring in-equality. This is connected to the issue of Szeg˝o asymptotics of OPRL[674] (see the discussion in Section 6 around (6.51) for the definitionsof OPRL and OPUC). This asymptotics says that for certain classes, { p n } ∞ n =0 , of OPRL whose spectral measure has essential support [ − , z ∈ D \ { } thatlim n →∞ z n p n ( z + z ) (8.15) exists and is not identically zero (which determines asymptotics of p n ( x ) for x / ∈ [ − , − , dρ ( x ) = f ( x ) dx, x ∈ [ − , 2] (8.16) (see [665, Section 13.1]). For such measures, early on it was realizedthe critical condition on such measures is (cid:90) − log f ( x )(4 − x ) − / dx > −∞ (8.17) a condition known as the Szeg˝o condition after an analog for OPUCused by Szeg˝o in [707]. If one thinks about Jacobi parameters ratherthan measures, it is natural to allow pure points outside [ − , ± 2. Inthis regard, the best possible result when Killip and I began our work(discussed below) was ? (cid:104) T8.3 (cid:105) ? Theorem 8.3 (Peherstorfer-Yuditskii [504]) . Let dρ be a probabilitymeasure on R which has a pure a.c. part on [ − , of the form (8.16) and additional pure points { E ± j } N ± j =1 on ± (2 , ∞ ) so that f obeys theSzeg˝o condition (8.17) and, in addition, (cid:88) j, ± (cid:113) | E ± j | − < ∞ (8.18) Then the associated OPRL, { p n } ∞ n =0 , obey Szeg˝o asymptotics (8.15) . Taking into account that ± b n ≡ , a n ≡ ∞ (cid:88) n =1 | a n − | + | b n | < ∞ ⇒ (8.17) [Nevai] (8.19) In my work with Killip [374], we proved that(8.18) + LHS of (8.19) ⇒ (8.17) [Killip-Simon] (8.20) It was then clear that a suitable critical Lieb-Thirring inequalityfor Jacobi matrices would prove Nevai’s conjecture. Fortunately, DirkHundertmark, one of the coauthors of [311] had just come to Caltechas a postdoc and we proved ? (cid:104) T8.3A (cid:105) ? Theorem 8.4 (Hundertmark-Simon [312]) . One has that (cid:88) j, ± (cid:113) ( E ± j ) − ≤ ∞ (cid:88) n =1 | b n | + 4 ∞ (cid:88) n =1 | a n − | (8.21) Indeed, mimicking the proof of [311] fairly easily proves (8.21) when a n ≡ ⇒ (8.18) and so (8.20) ⇒ (8.19)proving Nevai’s conjecture. [312] also proved general p > , ν = 1Lieb-Thirring and Bargmann bounds for Jacobi matrices.Over the next few years, with graduate students and postodcs, Iexplored various extensions. Zlatoˇs and I [691] proved a variant of [374]for oscillatory b n and a n − a n =1 + ( − n αn or b n = ( − n βn .It was widely believed in the OP community that if Szeg˝o asymp-totics holds, one must have a Szeg˝o condition so it came as a surprise WELVE TALES 93 when Damanik-Simon [123] found necessary and sufficient conditionsfor Szeg˝o asymptotics to hold that allowed many examples with Szeg˝oasymptotics where one has Szeg˝o asymptotics even though the Szeg˝ocondition and finiteness of the LHS of (8.18) fail; indeed, [123] hasexamples where the sum of ( | E ± j | − α is infinite for all α < .I was also involved in several projects leading to Lieb-Thirringbounds for perturbations of non-free Schr¨odinger operators and Jacobimatrices. Frank, Weidl and I [193], to quote our result for Schr¨odingeroperators, proved that if there is a solution, u , of ( − ∆ + V ) u = 0and c , c ∈ R so that 0 < c ≤ u ( x ) ≤ c < ∞ for all x , then Lieb-Thirring inequalities for perturbations of − ∆ imply them for perturba-tions of − ∆ + V (with adjusted constants). In particular, this impliesperturbations of periodic Schr¨odinger obey a Lieb-Thirring bound atthe bottom of the spectrum. A similar analysis gives such bounds forperturbations of periodic Jacobi matrices at the top and bottom of thespectrum and also for perturbations of almost periodic finite gap Jacobimatrices [693, 102]. These results depend on a ground state representa-tion for Schr¨odinger operators that goes back to Jacobi and which washeavily used in work in constructive quantum field theory. Somewhatsurprisingly, [193] seems to be the first place that this representationwas worked out for Jacobi matrices. This ground state representationwas used earlier to compare operators by Kirsch and me [379] in a pa-per, that, in particular, got interesting bounds on effective masses insolid state Hamiltonians.It was natural to ask about Lieb-Thirring bounds and the analog ofthe Nevai conjecture for eigenvalues in gaps of perturbations of peri-odic and finite gap almost periodic Jacobi matrices. For the periodicproblem with all gaps open, this accomplished by Damanik, Killip andme [122] and for finite gap problems by Frank and me [192] after partialresults by Birman [69] and Hundertmark-Simon [314].We saw that (8.10) was only proven for ν ≥ 3. There is a goodreason for this. A bound like(8.10) implies that for λ small, − ∆ + λV has no bound states (for say, all V ∈ C ∞ ) but it was known that for anegative square well, i.e. V the negative of the characteristic functionof ( − a, a ) in R or of a disk of radius a in R , − ∆ + λV has a negativeeigenvalue for all λ . I asked what happens for general V and proved[602] that ? (cid:104) T8.4 (cid:105) ? Theorem 8.5 (Simon [602]) . Let V be a real-valued function on R ,not identically , obeying (cid:90) (1 + | x | ) | V ( x ) | dx < ∞ (8.22) ? ? Then − d dx + λV ( x ) has a negative eigenvalue, E ( λ ) for all small, pos-itive λ if and only if (cid:90) V ( x ) dx ≤ and if that is the case, one has that α ( λ ) ≡ ( − E ( λ )) / = − λ (cid:90) V ( x ) dx − λ (cid:90) V ( x ) | x − y | V ( y ) dxdy + o ( λ )(8.24) Remarks. 1. This work was motivated by Murph Goldberger, who atthe time was department chair of physics at Princeton (I was Directorof Graduate Students) and who later was the President of Caltech atthe point when I was recruited. He had organized a group of particletheoretical physicists (Jason) who worked on DoD projects for a fewweeks each summer. While studying some problems on sound wavesin water (not quantum mechanics!), Murph with Henry Abarbanel andCurt Callen got interested in negative eigenvalues in one dimension andfound (8.24) as a formal series. Murph wanted to know if I could provesomething.2. For α to always be positive when (8.23) holds, one must havethat (cid:82) V ( x ) dx = 0 ⇒ (cid:82) V ( x ) | x − y | V ( y ) dxdy ≤ 0, a fact summarizedby saying the | x − y | is a conditionally negative definite kernel; this isindeed true.3. Blankenbecler, Goldberger and I [72], and independently Klaus[385], showed that (8.24) could be replaced by the weaker (cid:90) (1 + | x | ) | V ( x ) | dx < ∞ (8.25) 4. If V ( x ) ∼ − ax − β as x → ∞ with 1 < β < 2, something interest-ing happens. There are now infinitely many eigenvalues but most areO( λ / (2 − β ) ) while the lowest eigenvalue is O( λ / (3 − β ) ). This is provenin [72] which also has results when β = 2.5. Not only does (8.17) fail if ν = 1 , 2, but there is a result [605,Remark 3 on pg 315] that if (cid:107)·(cid:107) is any translation invariant norm ona vector space of functions that includes some non-zero, everywherenon-positive continuous functions of compact support, then for any N and any ε , there is a V with (cid:107) V (cid:107) < ε and so that − ∆ + V has at least N negative eigenvalues! WELVE TALES 95 6. We think of this result which violates a putative quasi-classicalbound as an example of non-quasi-classical eigenvalue behavior.7. [602] also proves that if V decays exponentially, then ( − E ( λ )) / is analytic in λ at λ = 0.8. [602] also has results when ν = 2. In that case, if (cid:82) V ( x ) d x < 0, we have a negative eigenvalue for − ∆ + λV but for small λ it isO(exp( − d/λ ))!The proof of the theorem is not hard. The one-dimensional Birman-Schwinger kernel for E = − α has the form K α ( x, y ) = | V | / ( x ) exp( − α | x − y | ) V / ( y ) (8.26) ? ? The Birman-Schwinger principle says that E is an eigenvalue of − d dx + λV if an only if − λ − is an eigenvalue of K α . In more thantwo dimensions, (cid:107) K α (cid:107) is bounded as α ↓ λ < sup (cid:107) K α (cid:107) , then − ∆ + λV has no negative eigenvalues, but in 1 (or 2) dimensions, (cid:107) K α (cid:107) diverges as α ↓ 0. The reason that there is only one negative eigen-value when λ is small (at least when (8.24) or (8.25) holds) is that thedivergent piece is rank one. To see this, [602] writes K α = L α + M α (8.27) ? ? L α ( x, y ) = | V | / ( x ) V / ( y ) / α (8.28) ? ? M α ( x, y ) = (2 α ) − | V | / ( x ) (cid:2) e α | x − y | − (cid:3) V / ( y ) (8.29) ? ? so lim α ↓ M α ≡ M exists where M ( x, y ) = | V | / ( x ) | x − y | V / ( y ) (8.30) ? ? Thus, the Birman-Schwinger principle is equivalent to α = − λ (cid:104) V / , (1 + λM α ) − | V | / (cid:105) (8.31) ? ? The leading term on the right is − λ (cid:82) V ( x ) dx and the next is λ (cid:104) V / , M | V | / (cid:105) .This work lead to several threads of later work. In [611], I asked whena new eigenvalue, E ( λ ), issuing from E = 0 at λ = λ is O( λ − λ ) andproved that (cid:104) T8.5 (cid:105) Theorem 8.6. Suppose that B is a relatively compact symmetric per-turbation of a self-adjoint operator, A , that σ ess ( A ) includes [0 , ε ] forsome ε > and that λ > is such that A + λB has exactly one morenegative eigenvalue (counting multiplicity) for λ ∈ ( λ , λ + δ ) thanfor λ ∈ ( λ − δ, λ ) . Then there is one eigenvalue, E ( λ ) , near for λ ∈ ( λ , λ + δ ) and lim λ ↓ λ E ( λ ) λ − λ = α (8.32) ? ? exists. Moreover, α (cid:54) = 0 if and only if A + λ B has eigenvalue and with that case, the eigenvalue is simple and α = (cid:104) ϕ, Bϕ (cid:105) where ( A + λ B ) ϕ = 0 with (cid:107) ϕ (cid:107) = 1 . In [623, 386], I made what turned out to be an important definition. Definition 8.7. Let ν ≥ V ∈ L p ( R ν ) ∩ L q ( R ν ) for some p < ν Theorem 8.8 ([633]) . Let V, W be two C ∞ functions on R ν so that(1) For some A, R > , one has V ( x ) ≥ A if | x | ≥ R (2) V ( x ) ≥ for all x (3) V ( x ) vanishes only at { x ( j ) } Mj =1 ( M ≥ ) and at each such min-imum, the matrix A ( j ) k(cid:96) = ∂V∂x k ∂x (cid:96) ( x ( j ) ) is strictly positive.(4) W is bounded from belowLet H ( j ) = − ∆ + (cid:80) k,(cid:96) A ( j ) k(cid:96) x k x (cid:96) + W ( x ( j ) ) . Let { e α } ∞ α =1 be an orderingof the union over j of the eigenvalues of H ( j ) (counting multiplicity) WELVE TALES 97 so that e ≤ e ≤ . . . . Then H ( λ ) ≡ − ∆ + λ V + λW (8.33) has eigenvalues, { E α ( λ ) } ∞ α =1 , with E ( λ ) ≤ E ( λ ) ≤ . . . at the bottomof its spectrum and for any α = 1 , , . . . we have that lim λ →∞ E α ( λ ) λ = e α (8.34) ? ? Moreover, each E α ( λ ) has an asymptotic series in λ − to all orders E α ( λ ) = e α λ + a (0) α + a (1) α λ − + . . . (8.35) ? ? I always thought of the paper in which this theorem appeared as EdWitten’s homework assignment because one motivation for this workwas his wonderful paper on the supersymmetric proof of Morse inequal-ities and the Morse index theorem [738]. In it, he used this theorem(or rather its generalization when functions on R ν are replaced by thetangent bundle on a compact manifold and − ∆ by a Laplace-Beltranioperator (also discussed in [633])). When using this result, Witten says Although the rigorous theory has apparently not been developed for op-erators acting on vector bundles on manifolds, the method used in Reedand Simon [527] , pp. 34–38, to treat the double well potential shouldsuffice with some elaboration for this case. In fact, the argument in[527] he refers to is one-dimensional and uses some other propertiesof the simple double well. The general one-dimensional case had beendone by Combes et al [107] but their arguments also depended on onedimension and are somewhat involved so I wrote my paper in part toget a proof that works in multiple dimensions and also one that is fairlysimple.[633] was the first paper of a series. The other papers dealt witheigenvalue splitting in the situation where multiple wells have the sameeigenvalue. It is simpler to discuss the case of the lowest eigenvalue as-suming a double degeneracy. A basic role is played by the Agmonmetric, mentioned in Section 6 (in the page after (6.37)), which wasknown to determine the rate of decay of eigenfunctions. In this situ-ation it is defined as the distance in the Riemann metric V ( x )( dx ) ,i.e. ρ ( x, y ) = inf (cid:26)(cid:90) (cid:112) V ( γ ( s )) | ˙ γ ( s ) | ds (cid:12)(cid:12)(cid:12)(cid:12) γ (0) = x, γ (1) = y (cid:27) (8.36) ? ? over all smooth paths, γ ( s ) , ≤ s ≤ 1, between x and y . The mainresult of [634, 637] is (cid:104) T8.7 (cid:105) Theorem 8.9 ([637]) . Let V be a C ∞ function on R ν (and W = 0 )obeying conditions (1)-(3) of Theorem 8.8. Suppose there are two points a (cid:54) = b where V vanishes and that e = e < e so that e and e areeigenvalues associated to the operators, H ( j ) , at the points a and b . Let j a (resp j b ) be characteristic functions of small balls about a (resp b ),balls that are so small that they are disjoint. Let Ω λ be the normalizedground state of the operator, H ( λ ) , of (8.33) . Suppose that lim inf (cid:107) j a Ω λ (cid:107)(cid:107) j b Ω λ (cid:107) > Then lim λ →∞ − λ − log [ E ( λ ) − E ( λ )] = ρ ( a, b ) (8.38) ? ? Remarks. 1. (8.37) says that the ground state lives near both minima.One condition that guarantees this is if there is a Euclidean rotationor reflection of order 2 that leaves V invariant with Ra = b . In thatcase the limit is exactly 1 / 4. That holds for the famous 1 D double wellwhere V ( x ) = x ( x − . In that case ρ is given by a WKB integraland this results was proven by various authors in the ten years beforemy result (see [637] for references) but my work was the first rigorousresult in more than one dimension.2. The proof controls eigenfunction decay by writing the eigenfunc-tions in terms of path integrals and using the method of large deviationsto single out a minimum action solution. It is a basic fact of classicalmechanics that minimum action is equivalent to minimum distance ina suitable metric.3. The importance of minimum action paths to leading order tun-nelling in mult-dimensions was noted in the theoretic physics literatureseveral years before my work. These solutions were called instantons ;see [637] for references to that literature.4. Shortly after my work, Helffer-Sj¨ostrand [283] developed a pow-erful microlocal analysis approach to these problems and recovered theresults of Theorems 8.8 and 8.9 that got higher order terms and workedin a more general setting as they discussed in a number of later papers.5. I wrote two later papers [638, 640] on some specialized situationsrelated to Theorem 8.9.Kirsch and I [378] proved an interesting universal tunnelling bound ? (cid:104) T8.7A (cid:105) ? Theorem 8.10 (Kirsch-Simon [378]) . Let V be a continuous func-tion on R so that − d dx + V ( x ) has discrete eigenvalues { E j } Nj =1 be-low any essential spectrum. Let n < N + 1 and suppose for some α > , we have that V ( x ) ≥ E n + α on R \ [ a, b ] . Let λ = WELVE TALES 99 max E ∈ [ E n − ,E n ]; x ∈ ( a,b ) (cid:112) | E − V ( x ) | . Then E n − E n − ≥ πλ α ( λ + α ) − e − λ ( b − a ) (8.39) ? ? The last topic that I discuss in this section concerns another non-quasi-classical situation. Just as Theorem 8.6 was motivated by a ques-tion posed to me by Goldberger and Theorem 8.8 by Witten, this workwas motivated by a query from some theoretical, non-mathematical,physicists. In this case, I was asked by Jeffrey Goldstone and RomanJackiw if the two dimensional Schr¨odinger operator H = − ∂ ∂x − ∂ ∂x + x y (8.40) has purely discrete spectrum or not. They noted that one was usedto the condition for purely discrete spectrum of − ∆ + V being that V ( x ) → ∞ as x → ∞ . While this failed for V ( x, y ) = x y since V vanished on the axes, they suspected the spectrum was discretesince it went to infinity in all but four directions. In fact, the naturalquasi-classical condition of finite phase space volume, i.e. |{ ( x, p ) | p + V ( x ) ≤ E }| < ∞ for all E also fails in this case. A closely relatedquestion involves the operator H = − ∆ Ω D ; Ω = { ( x, y ) ∈ R | | xy | ≤ } (8.41) Motivated by their question, in [635], I proved that (cid:104) T8.8 (cid:105) Theorem 8.11 ([635]) . The operators H of (8.40) and H of (8.41) both have purely discrete spectrum. Remarks. 1. [635] gives six proofs that H has purely discrete spec-trum. The simplest proves that H has purely discrete spectrum (andthat easily implies that so does H ) as follows: It follows by scaling andthat fact that − d dq + q has smallest eigenvalue 1 that − d dq + ω q ≥ | ω | which implies that − ∂ ∂x + x y ≥ | y | . Interchanging x and y , addingthe two and multiplying by 1 / H ≥ ( − ∆ + | x | + | y | ) ≡ H (8.42) ? ? Since H has purely discrete spectrum, by the min-max principle [680,Theorem 3.14.5], so does H . The “defect” in this proof is that it turnsout that H for large energies is much larger than H . The numberof eigenvalues of H larger than E grows like E (by a quasi-classicalestimates) while for H only like E / ln E (as discussed below) whichis much smaller.2. The operator H that Goldstone and Jackiw asked me to look atwas a toy model for a more involved model they were really interested 00 B. SIMON in. Let A be a semi-simple Lie algebra and let − ∆ A be the Laplacianin the inner product on A given by the negative of the Killing form.For ν ≥ 2, let A ν be the set of ν -tuples, ( A , . . . , A ν ), of elements of A .Then they were interested in the operator on L ( A ν ) H = − (cid:88) i ∆ A i − (cid:88) i There is also discussion in the literature of the analog of H whenDirichlet boundary conditions are replaced by Neumann boundary con-ditions. As I described in Section 6 around equation (6.41), if one looksat the Neumann Laplacian ofΩ = { ( x, y ) ∈ R | x > , | y | < f ( x ) } (8.46) ? ? then if the V of (6.40) goes to zero slowly, − ∆ Ω N has some a.c. spectrum.Evans and Harris [178] found necessary conditions on when this oper-ator has purely discrete spectrum. For many such operators, Jakˇsi´c,Molˇcanov and I [331] found the leading asymptotics for the numberof eigenvalues, N Ω N ( E ), of asymptotics as E → ∞ . In particular, wefound for the interesting case f ( x ) = exp( − x α ) that N Ω N ( E ) ∼ | Ω | E, if α > (cid:0) | Ω | + (cid:1) E, if α = 2 C α E / / (2( α − , if 1 < α < ? ? where C α = 14( α − √ π (cid:16) α (cid:17) / (1 − α ) Γ(1 / (2( α − / / (2( α − α > 2, we have a quasi-classical Weyl behavior, but for other alpha,we have non-quasi-classical behavior.A final remark on non-quasi-classical eigenvalue behavior. Kirschand I [380] (motivated again by a question from a non-mathematicalphysicist - in this case, Michael Cross) found such behavior forthe growth of the number of eigenvalues below E as E ↑ − ∆ + c (1 + | x | ) − .9. Almost Periodic and Ergodic Schr¨odinger Operators (cid:104) s9 (cid:105) In AY 1980-81, I visited Caltech as a Fairchild Distinguished Scholar(I got an offer during the year and stayed). I was looking forward toa year with no teaching, only one postdoc (Yosi Avron, who also hada leave from Princeton) and only one grad student (Peter Perry), ayear where I expected to be able to focus on research with few dis-tractions. I had the impression that many of the areas I had focusedon were winding down at least as far as my involvement. CQFT wasmainly using involved expansions and estimates, not my forte, andthe leap to four dimensional space-time which required going beyondsuperrenormalizable theories seemed daunting (and still hasn’t hap-pened!). The hottest open question in N -body NRQM was asymptoticcompleteness and, while I was hopeful the N -body Mourre estimatesthat I’d recently proven with Perry and Sigal (see Section 6) would 02 B. SIMON be useful, I had no plan for how to proceed. So I suggested to Yosithat we look at moving into a new area. There seemed to be two toconsider: Scr¨odinger operators with almost periodic potentials (whereI was aware of some interesting non-rigorous work of Aubry [20, 17])and quasi-classical eigenvalue counting (where I was aware of a thenrecent preprint of Helffer and Robert [282] - a kind of multidimensionalversion of Bohr-Sommerfeld quantization rules). They both seemed in-teresting and promising. After thinking about it, I said to Yosi: “Let’stry to do both. We’ll do almost periodic first - it doesn’t look verycomplicated or involved. We’ll finish it up in six months and thenwe can turn to quasi-classical”. Little did I realize! Almost periodicpotentials was a major focus of my work for more than 15 years and,now, almost 40 years later, while there has been remarkable progress,it is still an active area with its own separate conferences. While, inthe few years after that, I did some quasi-classical stuff (see Section8), I never worked on detailed eigenvalue locations and related issuesalthough it to has become an active area (see, for example, Zelditch[741] for a recent review of some aspects).Before turning to the details of this subject, I should point outthat it is intimately connected to the subject of Section 11 (randomSchr¨odinger operators) and connected to the subject of Section 12 (sin-gular spectrum) so some papers may only be mentioned here and dis-cussed in more detail in later sections. Moreover, our formal discussionbelow will start with the general framework of Schr¨odinger operatorsand Jacobi matrices with ergodic potentials which encompasses ran-dom and almost periodic potentials as special cases. We will be verybrief in this presentation referring the reader to the relevant sections ofmy book with Cycon et al. [117], the lovely review of Jitomirskaya [339]or, for more comprehensive discussion, the books of Carmona-Lacroix[91], Pastur-Figotin [499], Stollmann [700], or Aizenman-Warzel [13].My initial work, much of it with Avron [41, 44, 46, 57, 113, 114,630, 135] was a major focus of my research during the three year pe-riod 1981-1983 (which was extremely fruitful including also my earlywork on TKNN integers and Berry’s phase and their geometric signif-icance [36, 632] (see Section 10), my work on ultracontractivity [127](see Section 3), my work on multiwell problems [633, 637] and on non-classical eigenvalue asymptotics [635, 636] (see Section 8), my discoveryof localization for slowly decaying random potentials [628] (see Section11), the completion and publication of my influential review articleon Schrodinger semigroups [626] (with over 1400 citations on Google WELVE TALES 103 Scholar), several miscellaneous papers on NRQM with Coulomb po-tentials [65, 439] as well as the preparation of my 45 hour, Bayreuthlecture course in the summer of 1982 which turned into [117]).I gave a review talk on the early work on almost periodic Schr¨odingeroperators at the 1981 Berlin ICMP [627] which became known as thealmost periodic flu paper because I started the talk by remarking on thefact that there seemed to be a worldwide explosion of work in this newarea that I dubbed the almost periodic flu. Indeed, besides my work inCalifornia with Avron, there was work by Bellisard and collaboratorsin France (reviewed with lots of references in [58]; notable was his useof C ∗ -algebra methods), Chulaevsky [105] in Moscow and by Moser[476], Johnson-Moser [344] and Sarnak [550] in New York (notable wasthe Johnson-Moser invention of rotation number and the resulting gaplabelling).The basic framework is a probability measure space (Ω , Σ , µ ) withexpectation, E , a distinguished bounded function, f : Ω (cid:55)→ R , anda distinguished group g (cid:55)→ T g of ergodic measure preserving maps in-dexed by the reals or the integers (see [679, Sections 2.6-2.9] for more onthe ergodic and subadditive ergodic theorems). In the continuous case,one considers a potential V ω ( x ) = f ( T x ( ω )) and ergodic Schr¨odingeroperator H ω = − d dx + V ω ( x ) acting on L ( R ) and in the discrete case,one takes diagonal elements b n ( ω ) = f ( T n ( ω )) and ergodic discreteSchr¨odinger operator( H ω u ) n = u n +1 + u n − + b n ( ω ) u n (9.1) acting on (cid:96) ( Z ). While this is the simplest example, one often general-izes (and we will occasionally below) in three ways: one can allow suit-able unbounded f (cid:48) s (often bounded from below), one can replace R or Z by R ν or Z ν with the multidimensional Laplacians, and, finally, one canconsider ergodic Jacobi matrices rather than only discrete Schr¨odingeroperators (i.e. allow ergodic a n ’s).Two special cases are the random (discussed mainly in Section 11)and almost periodic cases (the latter is the subject of this section aftera general discussion of some common objects). For the discrete randomcase, Ω = [ a, b ] Z , f ( { ω j } j ∈ Z ) = ω , T ( { ω j } j ∈ Z ) = { ω j +1 } j ∈ Z , and dµ ( { ω j } j ∈ Z ) = ⊗ j ∈ Z dκ ( ω j ) (9.2) so b j ( ω ) is a sequence of independent identically distributed randomvariables (aka, iidrv). The special case where dκ is uniform distribu-tion on an interval is usually called the Anderson model (after [16] for 04 B. SIMON which Anderson got the Nobel prize for claiming the model had local-ized states as we’ll discuss in Section 11). We’ll sometimes call the gen-eral iidrv case the generalized Anderson model . One sometimes studiesunbounded distributions or even non-independence but demands the b j be “really random”, at least defined by a Markov process. Since Inever worked directly on continuum random operators, I’ll leave thedescription of those models to the books mentioned above, especiallyAizenman-Warzel [13] and Stollmann [700]. I do however note that toaccommodate models with say a fixed potential localized about latticepoints in R ν with iidrv coupling constants, one needs to modify theset up to only require ergodicity under a discrete group even for R ν models.The other case is almost periodic functions. (For more on the generaltheory of almost periodic functions, see, e.g. [680, Section 6.6]). In thiscase, Ω is a separable, compact, abelian group, called the hull , there isa homomorphism S : G → Ω (with G = R or Z ) so that T g ( ω ) = ωS ( g )and f : Ω → R . Two important special cases are where S is a windingline on a finite dimensional torus, Ω, viewed as a product of copiesof ∂ D with complex product as the group product, in which case thepotential is called quasiperiodic and the case where the potential is auniform limit of periodic functions of longer and longer commensurateperiods (e.g. V ( x ) = (cid:80) ∞ n =1 − n cos( x/ n )) in which case the potentialis limit periodic . The most famous example is( H α,λ,θ u ) n = u n +1 + u n − + λ cos( παn + θ ) u n (9.3) the almost Mathieu operator (henceforth AMO). In much recent liter-ature, what I call λ is called 2 λ (so the self dual point is λ = 1) but I’llfollow the convention of the older literature. I like to joke that therehave been more papers in the Annals of Mathematics about the AMOthan about any other single mathematical object. In the physics liter-ature, this is called Harper’s equation when λ = 2 and arose as a tightbinding approximation to a 2D electron in magnetic field ( α is thenthe magnetic flux per unit cell). The name almost Mathieu equationis one I introduced in [46] and [627]. I took it from the fact that thedifferential equation − d udx + λ cos( x ) u ( x ) = Eu ( x ) (9.4) is called the Mathieu equation (with Avron [42], I had then recentlystudied the asymptotics of its gap widths as E → ∞ ). My name is ajoke based on the fact that (9.3) is almost (9.4) and is also only almostperiodic if α is irrational (while (9.4) is periodic). WELVE TALES 105 Two basic objects associated to one dimensional ergodic operatorsare the density of states (DOS) and the Lyaponov exponent. The DOS,unlike the Lyaponov exponent, makes sense in higher dimensions, but,for simplicity, let us mainly focus on the one dimensional discrete case.For each ω , H ω defines a self-adjoint operator on (cid:96) ( Z ), and so definesa spectral measure, dµ ω ( E ), defined by (cid:90) f ( E ) dµ ω ( E ) = (cid:104) δ , f ( H ) δ (cid:105) (9.5) ? ? The DOS measure dk ( E ) is defined by dk = E ( dµ ω ) (9.6) The integrated density of states ( IDS ) is then defined by k ( E ) = dk (( −∞ , E )) (9.7) ? ? If χ L is the characteristic function of { n | − L ≤ n ≤ L } , then trans-lation covariance shows that if P B ( H ω ) is the spectral projection for aset, B , then (2 L + 1) − E (Tr( χ L P B ( H ω ) χ L )) = (cid:90) B dk ( E ) (9.8) ? ? This together with translation covariance and the Birkhoff ergodic the-orem [679, Theorem 2.6.9] imply that for a.e. ω and all continuousfunctions f on R of compact support one has thatlim L →∞ (2 L + 1) − Tr( χ L f ( H ω ) χ L )) = (cid:90) f ( E ) dk ( E ) (9.9) A simple argument (e.g. restricting to moments) then shows that dk isalso the limit of the eigenvalue density of H ω restricted to large boxeswith either periodic or Dirichlet boundary conditions.The earliest mathematical work on the DOS was by Benderski˘i-Pastur [62] who defined it in the random case as a limit of box eigen-value counting. See [117, pg. 175] for additional references on work onthe random case prior to Avron-Simon [46] whose principle theme wasthe DOS for the almost periodic case (and, as we’ll discuss below, theThouless formula, Aubry duality and spectral properties of the AMO).We introduced the definition via (9.6) and the formula (9.9) which weproved held for every ω in the hull, rather than just almost every ω . Wealso proved the equality to the definition via (9.6) to the definition viaperiodic or Dirichlet boundary conditions and also (up to a factor of π )to the then recently defined rotation number of Johnson-Moser [344].This equality under boundary conditions was natural given the statisti-cal mechanical analogy and the proof was not hard. Twenty-five years 06 B. SIMON later, its analog turned out to be very useful to prove a result [673] inthe theory of orthogonal polynomials that surprised many experts inthat subject.We also proved that for any ω , the spectrum of H ω is equal to thesupport of the measure dk and, in particular, spec( H ω ) is the same forall ω in the hull. We also proved that k was continuous in 1D andnoted the importance of continuity since that implies that for every E one has that (here H L,Dω is the restriction of the Hamiltonian H ω to thebox { n | − L ≤ n ≤ L } with Dirichlet boundary conditions which onecould replace by periodic boundary conditions). k ( E ) = lim L →∞ (2 L + 1) − H L,Dω ≤ E (9.10) ? ? Our proof of the continuity of k in 1D depended on the fact that in1D eigenvalues have multiplicity at most 1 (all that mattered was thefiniteness) so we suggested, but couldn’t prove, that k was continuous ingeneral dimension. Moreover, the proof only showed that lim ε ↓ [ k ( E + iε ) − k ( E − ε )] = 0 (which implies continuity for monotone functions)with nothing about how small differences are. Later, Craig and I [113]proved log H¨oloder continuity (using subharmonic function methodsintroduced in the subject by Herman [294]) and then extended this toany dimension [114] to get the continuity in any dimension that [46] hadconjectured. Since the proofs of [113, 114] use the Thouless formula,I’ll discuss it below. I note that shortly after us, Delyon-Souillard [151]found a distinct, really short, proof of the multidimensional continuity(but not log H¨oloder continuity).Before leaving the DOS, I should mention gap labelling, not becauseI contributed to it, but because it will be relevant to the discussionof Cantor spectrum and the ten martini problem. If H is a discreteSchr¨odinger operator of period L , the usual Bloch wave analysis showsthe spectrum can have up to L − /L . Johnson-Moser [344] (once one has the equality oftheir rotation number and the IDS) found an analog for the 1D almostperiodic continuum case (their method was extended to the discretecase by Delyon-Souillard [150]). Independently, Bellisard [54], provedthe same result using C ∗ -algebra methods, eventually using the sameidea for certain higher dimensional operators [56]. For any almostperiodic function, f , its average, A ( f ) = lim R →∞ (2 R ) − (cid:82) R − R f ( x ) dx is easily shown to exist. Given a real, ω , f is said to have a non-zeroFourier coefficient at ω if and only if A ( e − πixω f ) (cid:54) = 0. Since in the casethat x ∈ Z , these Fourier coefficients only depend on the fractional part WELVE TALES 107 of ω , we view ω as an element of R / Z which we write as [0 , 1) and addmod Z . One can prove that the set of ω with non-zero coefficient is acountable set. The set of reals which are finite sums and differences ofthose ω with non-zero coefficient is called the frequency module of f . Itis easy to see that f is quasiperiodic if and only if the frequency moduleis finitely generated. Moreover, unless the potentials are periodic, thefrequency module is dense in [0 , 1) or [0 , ∞ ). Gap labelling is the assertion that in any gap of the spectrum, thevalue of the IDS is a number in the frequency module. What this saysin case V or a, b are periodic where the frequency module is multiplesof 1 /p (with p the “true” period) is that the constant value of the IDSin a gap is among j/p, j = 1 , . . . (where in the discrete case j runsthrough p − open for a given j if thereis such an interval of constancy and closed if there is a single energywith k ( E ) = j/p . Earlier in 1976, I had proven in [603] that in thecontinuum case for a dense G δ of V ’s of period one, that all allowedgaps are open. For the discrete case there is a much more preciseanalysis [160, 466] that shows the set of period p Jacobi parameterswith at least one gap closed is a finite union of closed varieties withcodimension 2 so the set with all gaps open is an open set that is muchmore than generic. A little thought shows that if the all allowed gapsare open in the almost periodic case, the set of gaps is dense so thatthe spectrum of H is a Cantor set (i.e. a closed, perfect, nowhere denseset). Of course, it can be a Cantor set even if only many, rather thanall, gaps are open.Avron and I were struck by this Cantor spectrum. Just as we weredoing this work, pictures appeared from the Voyager flyby of Saturnwhich showed many more gaps in that planet’s rings than previousknown, so many that it almost appeared that the rings were nowheredense! We wrote a speculative paper [43] suggesting the structuremight be due to an almost periodic Hill equation although we pointedout that naive perturbation estimates of gap size produced were toosmall by several orders of magnitude so there would need to be somethen not understood phenomena increasing this gap size. Alas, theredoes not seem to be such a phenomenon and nature choose a differentmechanism.At this time, Avron and I [44], Chulaevski˘i [105], Moser [476] andPastur-Tkachenko [500] independently found classes of limit periodicSchr¨odinger operators with Cantor spectrum. We also proved the spec-trum remained purely absolutely continuous so the spectrum was apositive measure Cantor set. We also discovered that such absolutely 08 B. SIMON continuous spectrum still had all states with slow decay leading us todevelop a refinement of a.c. spectrum [41] .Mark Kac had moved to USC about the time I was visiting Caltechand at lunch one day in 1981, he and I discussed Cantor spectrum andthe AMO. We agreed that it was an interesting conjecture to prove thatthe operator H λ,α,θ of (9.3) had a Cantor spectrum for all irrational α and λ (cid:54) = 0 (if α is irrational, it is known (Avron-Simon [46]) that thespectrum is θ independent). “That’s a grand conjecture”, said Mark,“I’ll offer ten Martinis for its solution.” He later repeated this offerat an AMS meeting and I popularized it as the ten Martini problem.Added to the interest was the famous Hofstader butterfly [300, 301],a picture (see Figure 2) showing the spectrum at the critical value λ = 2 of the spectrum as a function of α (computed numerically for α = p/q with q not too large) which looks like a fractal. The tenmartini problem was solved in full in 2004 by Avila-Jitomirskaya [22](mentioned in Avila’s Fields Medal citation) after an important partialresult by Puig [515]. This is weaker than the result that all gaps areopen, something known as the dry form of the ten Martini problem(still partially open). Figure 2. The Hofstader ButterflyA year after my lunch with Kac, Bellisard and I [57] used the strat-egy of my periodic result [603]. We first proved that if α = p/q isrational and qθ is not a multiple of π , then all gaps were open (i.e. thespectrum has q − k ( E ) from Avron-Simon [46], themagic of the Baire category theorem showed that for a Baire genericset of ( α, λ ), the spectrum is a Cantor set! It is remarkable that withone’s Baire hands one can learn something about the irrational case WELVE TALES 109 (Cantor spectrum) by knowing something about the rational case eventhough, of course, in the rational case, the spectrum is never Cantor.When I told Mark about this on the phone admitting it wasn’t thefull result, he remarked “But it is still interesting! I’ll give you threemartinis for it.” So I always think of this as the three Martini result.Alas, before we met again, Mark was dead of abdominal cancer (thesame disease that felled the other half of the Feynman-Kac formula nottoo long afterwards).Returning to my basic series with Avron, I need to define theLyaponov exponent. I’ll do it first for the discrete case. Given a pairof potential Jacobi parameters, a > , b ∈ R and z ∈ C , one definesthe single step transfer matrix: A ( a, b ; z ) = 1 a (cid:18) z − b − a (cid:19) (9.11) so the difference equation a n u n +1 + b n u n + a n − u n − = zu n (9.12) is equivalent to ( u n +1 a n u n ) t = A ( a n , b n ; z )( u n a n − u n − ) t . I learned thetrick of putting a factor of a in the lower component which yields an A with det( A ) = 1 from Killip in about 2000 and it didn’t appear in theearlier papers.One defines the transfer matrix T n ( { a j , b j } nj =1 ; z ) = A ( a n , b n ; z ) A ( a n − , b n − ; z ) . . . A ( a , b ; z ) (9.13) ? ? We use T n ( z ; ω ) for the transfer matrix with a n ( ω ) , b n ( ω ). TheFurstenberg-Kesten theorem [679, Theorem 2.9.1 ] then implies thatfor each fixed z , for a.e. ω , one has that the Lyaponov exponent γ ( z ) = lim n →∞ n − (cid:107) T n ( z ; ω ) (cid:107) (9.14) exists and is a.e. ω independent. More can be proven: The Multi-plicative Ergodic Theorem [679, Theroem 2.9.10] says that for each z and a.e. ω , not only does (9.14) hold but there is a one dimensionalsubspace V z ; ω ⊂ C so thatlim n →∞ n − (cid:107) T n ( z ; ω ) v (cid:107) = (cid:26) − γ ( z ) , if v ∈ V \ { } γ ( z ) , if v ∈ C \ V (9.15) ? ? Thus for such z, ω , if γ > 0, then all solutions of (9.12) on a half lineeither grow or decay exponentially. We emphasize that the need fora.e. ω rather than every is not a mere technicality but, as we will see(in the discussion two paragraphs prior to (9.24)), can have dramaticconsequences. 10 B. SIMON An important role is played by what is called the Thouless formula : γ ( E ) = (cid:90) log | E − E (cid:48) | dk ( E ) (9.16) which relates the Lyaponov exponent, γ , to the IDS, k ( E ) in the dis-crete case. This is the form for the discrete Schr¨odinger case where a n ≡ 1; in general, one has an extra term E ( − log( a ( ω )) with theadded condition that this expectation is finite. It has the name be-cause of the 1974 work of Thouless [716] although it appeared earlierin the physics literature in a paper of Herbert and Jones [288]. In fact,closely related ideas, although not the exact formula go back to Szeg˝oin 1924 [708] who realized an important connection to two dimensionalpotential theory (for discussion of the basics of potential theory, see[285, 464, 520] or [679, Sections 3.6-3.7]) since the right side of (9.12)is the (negative of the) logarithmic potential of dk . I was not awareof this related work from the OP community in the 1980’s but onlymany years later at which point I wrote a summary article [670] thatexplained the use of potential theory ideas to spectral theorists and theopposite direction to the OP community as well as some new insights.Thouless’ basic idea is that the elements of the transfer matrix whenall a n ≡ P n ( z ), whose zeros are the eigenval-ues of the Hamiltonian in a box. Since log( P n ( z )) = (cid:80) nj =1 log | z − E j | ,where the sum is over the eigenvalues, (9.16) then follows from thefact that dk is the limit of the density of eigenvalues in a box. Avronand I realized this argument worked flawlessly when z lay outside the(convex hull) of the spectrum of H , but because of infinities in the logwas problematic for z on the real axis. Indeed, we noted in the almostperiodic case for z off the real axis, it held for all ω rather than just a.e. ω . In [46], we were able to use the fact that the integral on the rightside of (9.16) is the Hilbert transform of k ( E ) and the L continuity ofHilbert transform to prove that (9.16) holds for Lebesgue a.e. E in R and this suffices for some applications we made.Slightly later, Craig and I [113] were able to prove (9.16) for all E ∈ R . The key was the observation of Herman [294] that the limit in(9.14) was subharmonic. Since the integral on the right side of (9.16)is also subharmonic and, by Thouless’ argument, the equation holdsfor z non-real, it hold for all z by a regularity result on subharmonicfunctions. Craig and I realized that in general for fixed ω , the quantitylim sup n →∞ n − (cid:107) T n ( z ; ω ) (cid:107) might not be upper semicontinuous whichimplied that this lim sup might only have the right side of (9.16) as anupper bound. We also realized that since γ ( E ) ≥ 0, the measure dk in(9.16) can’t give too great weight to small sets which implied the log WELVE TALES 111 H¨older continuity. By looking at averages of positive Lyaponov expo-nents on strips, we could even extend the continuity result to higherdimension.Avron-Simon [46] also began the study of a fascinating subject, thepossible ω dependence of spectral components. Recall [680, Theorem5.1.12] that one can refine the spectrum into pure point, a.c. ands.c. pieces. It is a theorem of Kunz-Souillard [402] that these spectralpieces are a.e. constant in the general ergodic case (one might thinkthis is obvious by the ergodic theorem but the subtlety is proving themeasurability in ω of the projections onto various spectral pieces). Asmentioned above, Avron and I proved a.e. could be replaced by all inthe almost periodic case for the spectrum but as we’ll see shortly, thatis not true for two of the three spectral components.For the AMO, (9.3), there are interesting dependencies of spectraltypes on the coupling constant and frequency. A key aspect is whatis called Aubry duality [20, 17]. Formally, the Fourier transform maps H α,λ to λH α, /λ since it maps the finite difference operator to multi-plication by cos and turns multiplication by cos into a finite differenceoperator. Of course, this can only be formal since Fourier transform in (cid:96) ( Z ) maps not to itself but to L ( ∂ D , dθ/ π )! Aubry duality says thatthe IDS, k ( α, λ ; E ), of H α,λ obeys k ( α, λ ; E ) = k ( α, /λ ; 2 E/λ ) (9.17) One way of understanding the dual relation is to view the direct integralof H α,λ,θ over θ as an operator on L ( R ) and apply the appropriateFourier transform. Alternatively, following [17], one looks at α = p/q with θ = 0 on a set with q points on which finite Fourier transformmaps (cid:96) ( Z q ) to itself. One obtains (9.17) by approximating irrational α by rationals. Aubry-Andr´e’s argument [17] for the limit was formal;Avron and I [46] proved the necessary continuity (which only holds atthe irrationals!) to get the first rigorous proof of (9.17). Two immediateconsequences of Aubry duality are (here α is irrational)spec( H α,λ ) = λ H α, /λ ) (9.18) ? ? γ ( λ, α ; E ) = γ (4 /λ, α ; 2 E/λ ) + log( λ/ 2) (9.19) Aubry-Andr´e [17] have a number of conjectures about the almostMatthieu equation which I (and others) made some progress on in mywork. The first involves the conjecture about the Lebesgue measure, | spec( H α,λ ) | , of the spectrum | spec( H α,λ ) | = 2 | − | λ || (9.20) 12 B. SIMON based on numeric calculations. We note this implies zero Lebesguemeasure when λ = 2 (which has led to a lot of literature on what theHausdorff measure is of the set in that case; [341] has a summary ofsome of that literature as well as new results), an earlier conjecture ofHofstader [300, 301].In [48], Avron, von Mouche and I attempted to prove (9.20) andproved the equality for rational α if the left side is the Lebesgue mea-sure of the intersection over θ of spec( H α,λ,θ ) and proved the conver-gence of the Lebesgue measure of the union over θ as any sequence ofrationals approaches an irrational α . This implies that the left side of(9.20) is ≥ the right side for α irrational. The techniques of [48] havebeen used in many later works studying this problem. For α whosecontinued fraction expansions are not bounded, Last [412, 413] provedthe complete (9.20) for all λ . The set of α with bounded integers intheir continued fraction expansion is easily seen to have Lebesgue mea-sure zero and to be a nowhere dense F σ so Last’s result covers “most”irrationals but not all and, in particular, it does not cover the goldenmean which has been used in many numeric calculations. The result isnow known for all irrational α through a series of papers. The historyis reviewed in Jitomirskaya-Krasovsky [341] which has a simple proofof the general result.The other conjecture in [17] concerns spectral types. One starts with(9.19) and the fact that always γ ≥ λ > γ ( E ) ≥ log( λ/ 2) (9.21) ? ? (proven rigorously by Avron-Simon [46] and Herman [294]) whichlater was proven to hold with equality on the spectrum by Bourgain-Jitomirskaya [76]. By a result of Pastur [498] and Ishii [323], strictpositivity of the Lyaponov exponent implies that the spectrum therehas no a.c. component so [17] suggested that when λ > 2, the spectrumis pure point. The Fourier transform of a rapidly decaying eigenfunc-tion looks like a plane wave so their conjecture on spectral type wasa.c. spectrum when 0 < λ < λ > λ = 2 would be subtle and suggested perhaps therewould be eigenfunctions with power law decay.When Avron and I started thinking about the issue of spectral typefor the AMO, Peter Sarnak, then a graduate student, suggested to methat spectral properties might depend on the Diophantine propertiesof the irrational frequencies (see also [550]), that is how well thoseirrationals are approximated by rationals. The Liouville numbers are WELVE TALES 113 those irrationals α for which there exist rationals p k /q k with (cid:12)(cid:12)(cid:12)(cid:12) α − p k q k (cid:12)(cid:12)(cid:12)(cid:12) ≤ k − q k (9.22) ? ? while we say that ( α , . . . , α (cid:96) ) have typical Diophantine properties ifthere exist C and k so that for all integers, not all zero, we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:96) (cid:88) j =1 n j α j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C ( n + · · · + n (cid:96) ) − k (9.23) We say that α is Diophantine if (9.23) holds with (cid:96) = 1 and α = α . Itis well known that the Diophantine rationals in [0 , 1] have full Lebesguemeasure while the disjoint set of Liouville numbers is a dense G δ (pro-viding an interesting demonstration that the two notions of generic aredistinct). Avron and I decided that the picture of Aubry-Andr´e waslikely wrong when α was a Liouville number. I visited Moscow in thespring of 1981 and explained our expectation and Molchanov came upto me with a young mathematician, Sasha Gordon, who had shown[245] that if a potential, V , is well enough approximated by periodicpotentials, then − d dx + V on L ( R , dx ) has no square integrable eigen-functions. Avron and I found the easy extension to the discrete caseand used it and the Pastur-Ishii result to prove [45, 46] that when α is a Liouville number and λ > 2, then the AMO, H α,λ,θ , has purelysingular continuous spectrum for all θ . This was only one of the timesthat Gordon had a significant impact on my work - the other most sig-nificant one was his impact on my work on generic singular continuousspectrum (see Section 12). We also had two joint papers [248, 249]. Ithink Gordon, who was a very inventive mathematician, never got thecredit he deserved and I felt guilty that I might have benefitted fromhis brilliance as much as he did!The result with Avron on examples with purely singular spectrumand γ > θ , one has that for a.e. E ∈ R , every solutionof the difference equation H α,λ,θ u = Eu either decays or grows expo-nentially at both ±∞ . Gordon’s lemma implies that in the Liouvillecase, the solutions decaying in one direction can’t in the other so thespectral measures of H α,λ,θ live on the sets where Lyaponov behaviorfails to hold. The moral is that the a.e. set of θ where (9.14) mightnot hold, not only exists but can where be all the important stuff ishappening! 14 B. SIMON For Diophantine α , with λ > 2, the Aubry-Andr´e conjecture wasproven by Jitomirskaya [338] who proved for such values of the param-eters, one has dense point spectrum of H α,λ,θ for Lebesgue a.e. θ . Itis not a limitation that the proof is only for a.e. θ since the spectrumis purely singular continuous for a dense G δ of θ (see the discussion ofmy work with Jitomirskaya [343] in the next paragraph). That leave α which is neither a Liouville number nor Diophantine, a non-empty,uncountable, set of irrationals that is both of Lebesgue measure zeroand a subset of a nowhere dense F σ (so, in a sense, rare). For such α one looks at the continued fraction approximations p n /q n which are,the best rational approximations [677, Section 7.5], and defines β ( α ) ≡ lim sup n →∞ (cid:18) log q n +1 q n (cid:19) (9.24) (This measure of approximation in the context of almost periodicSchr¨odinger operators goes back to my paper on the Maryland model[639] in an equivalent form β ( α ) ≡ lim sup n →∞ − n − log( | sin( παn ) | ).Diophantine α have β = 0 and Liouville α have β = ∞ . Avila-You-Zhou [24] proved that if 2 < λ < e β ( α ) the spectrum of H α,λ,θ is purelysingular continuous for all θ and if λ > e β ( α ) , then the spectrum of H α,λ,θ is dense pure point for a.e. θ . See Jitomirskaya-Liu [342] formore on this case including a review of the literature and a detailedanalysis of the eigenfunctions.One of the results of the singular continuous revolution that I’ll dis-cuss in Section 12 is that Jitomirskaya and I [343] proved using Gor-don’s lemma that if a n ( ω ) = 1 and b n ( ω ) is an even almost periodicfunction, then for a dense G δ of ω in the hull, h ω has no eigenvalues.If it is a model where γ > ω . Since there are models (like AMO for λ > α Diophantine) where it is known there is dense point spectrumfor a.e. ω , we see there are examples where neither the point spectrumnor the s.c. spectrum are ω independent even though we know thatthey are a.e. ω independent. However, independently, Kotani [396] andLast and I [415] showed that a.c. spectrum is always ω independent.I later wrote a paper with Hof and Knill [298] in which we proved,using a relative of the ideas in [343] that certain weakly almost periodicpotentials taking only finitely many values (which are known to haveno a.c. spectrum [395]) have purely s.c. spectrum for a dense G δ setin their hull.Next, I turn to AMO when 0 < λ < 2. The results on point spectrumwith exponentially decaying eigenfunctions for α Diophantine and λ > WELVE TALES 115 α is Diophantine and λ < . For several years after [46], it was assumedthat the dual of singular continuous spectrum must be purely singularcontinuous, so it came as a big surprise when Last [412] proved that for all irrational α on has that σ ac ( H α,λ,θ ) ≥ − | λ | . At the time, Last wasa graduate student of Avron at Technion and Avron told me of Last’sresult. I assured Avron that I was sure Last was wrong. Since I wouldbe coming soon to Israel, rather than plow through the paper and figureout the error, I suggested we meet so I could determine where the errorwas and tell him! To my surprise, Last convinced me that his proofwas correct and that he could prove a kind of lower semicontinuity on | σ ac ( H ) | and then use the fact that Avron, van Mouche and I [48] hadproven the inequality for rational α . Once the blinders were removed, Irealized that my work then in progress with Gesztesy [219] provided anew proof of Last’s result! Indeed, we could slightly improve his resultsince where he had | σ ac | , we could obtain the potentially smaller | Σ ac | (Σ ac is the essentially support of the ac spectrum, that is the minimalclass of sets mod sets of Lebesgue measure 0 that supports all the a.c.spectral measures). We proved that ? (cid:104) T9.1 (cid:105) ? Theorem 9.1 (Gesztesy-Simon [219]) . Let H [ n ] be a sequence of pe-riodic discrete Schr¨odinger operators so that for each fixed m , b [ n ] m haslimit b m and let H be the discrete Schr¨odinger operator with potential b . Then for any open interval ( α, β ) ⊂ R , we have that | ( α, β ) ∩ Σ ac ( H ) | ≥ lim sup n →∞ | ( α, β ) ∩ Σ ac ( H [ n ] ) | (9.25) ? ? For AMO with | λ | < 2, this leaves the question of the point and sin-gular continuous spectra which were expected to be empty and provingthat first for a.e. θ and then for all θ was open for many years; indeed,proving this for non-diophantine α (Jitomirskaya [338] had handled theDiophantine case for a.e. θ ) was one of the list of problems I sent to the2000 ICMP [658]. The full result was settled by Avila [21] (It is mostunfortunate that this paper has never appeared. It seems the blame isshared by the top journal that rejected it and by the author who thenrefused to send it elsewhere).Finally, in our discussion of AMO, I mention the self dual point | λ | = 2 which is often quite subtle. In [249], Gordon, Jitomirskaya,Last and I proved that if the spectrum of H α,λ,θ has zero measure for λ = 2 and some irrational α (the spectrum is θ independent), then fora.e. θ the spectrum is purely singular continuous. At the time the zeromeasure result was known for most, but not all, irrational α but, as justmentioned, it is now known for all irrational α . That left the question 16 B. SIMON of whether there might be an exceptional set with some (or even all)point spectrum. Very recently, Jitomirskaya [340] proved there are nopoint eigenvalues for any α and any θ . This paper includes a discussionof earlier work between the 1997 paper of Gordon et al [249] and her2020 breakthrough.That concludes our survey of the refined spectral analysis of AMOand I conclude this section with a summary of some of my other paperson almost periodic operators.(1) Kotani Theory . In 1982, I received a brilliant paper by S. Kotani[393] which dealt with ergodic continuum 1D Schr¨odinger operators.Since it dealt with a.c. spectrum, it was mainly of interest for thealmost periodic case. It had three main results(a) A kind of converse of the Pastur-Ishii theorem, namely, if γ ( E ) =0 on a Borel subset A ⊂ R of positive Lebesgue measure, then for a.e. ω , H ω has a.c. spectrum on A .(b) If γ ( E ) = 0 on an open interval I ⊂ R , then the spectrum ispurely a.c. on I .(c) If γ ( E ) = 0 on a Borel subset A ⊂ R of positive Lebesguemeasure, then, the process x (cid:55)→ V ω ( x ) is deterministic which meansthere is no a.c. spectrum in “truly random” cases.I discovered it was not so straightforward to extend this to the caseof discrete Schr¨odinger operators but I succeeded in [630] which wasused by many later authors. In [398], Kotani and I joined forces toextend these results discrete strips.(2) Deift-Simon Theory I wrote a paper with Deift [135] that focusedon aspects of a.c. spectrum motivated by Kotani [393] and some resultsof Moser [476] who had proven that the rotation number, α ( E ) = πk ( E ), obeys dα ( E ) /dE ≥ on the spectrum of periodic continuum Schrodinger operators and alsofor the particular limit periodic potentials he studies in [476]. In[135], we noted that (9.26) could not hold in general for all ergodicSchr¨odinger operators, essentially because of the phenomena of Lifshitztails (see Section 11) but we proved in the continuum case, it holds onthe set where γ ( E ) = 0 and for the discrete Schr¨odinger operator (i.e.Jacobi matrix with a ≡ 1) one has the stronger2 sin( α ) dα ( E ) /dE ≥ ? ? on A ≡ { E | γ ( E ) = 0 } which implies that | A | ≤ 4. We also provedthat the a.c. spectrum has multiplicity 2. While we regarded these asthe most important results in the paper (as seen by our abstract), thispaper is probably best known for two more technical aspects. First we WELVE TALES 117 construct L (in ω ) eigenfunctions for energies in the a.c. spectrum,which, for example, plays a critical role in the work in (4) below. Sec-ondly, there is a claim in [249] that our results imply mutual singularityof the singular parts of the spectral measures for a.e. pair ( ω, ω (cid:48) ) basedon a theorem in [135] that for every real E , a certain set of ω associatedto the singular spectrum has measure zero. Unfortunately, the theoremin [135] is sloppily stated in that in the section it appears, there is animplicit condition that γ > λ where γ = 0, that paper has one incorrect claim. Inote that the claimed mutually singularity when γ = 0 is still openalthough many expect that it is true.(3) The Maryland Model Dick Prange and two of his postdocs atMaryland found and studied [186, 260, 187, 514] a fascinating exactlysolvable almost periodic model which I dubbed “the Maryland model”,a name that has stuck in later literature. I wrote two papers [631, 639]on rigorous aspects of the model which had some overlap with anotherindependent rigorous analysis by Figotin-Pastur [185]. The model isjust like AMO but cos is replaced by tan, i.e.( H α,λ,θ u ) n = u n +1 + u n − + λ tan( παn + θ ) u n (9.28) ? ? Since tan is unbounded, one needs to eliminate the countable set of θ ’swhere for some n , παn + θ is of the form ( k + ) π ; k ∈ Z . Then onehas a well defined but unbounded operator. When α is Diophantine,The Maryland group found an explicit set of eigenfunctions but didn’tprove it complete and their computation of the density of states hadother formal elements.One surprise is that the DOS was the same as for the Lloyd modelwhich is the random model whose single site distribution is π − λλ + x dx which physicists call Lorentzian and mathematicians the Cauchy distri-bution. λ is called the half-width of the distribution. This distributionhas the following weird property. If X , X are two independent ran-dom variables each with a Cauchy distribution of the same half-width,then for any t ∈ [0 , tX + (1 − t ) X is also a Cauchy random variablewith half width λ . [631] noted that this implies that the Lloyd modelhas the same DOS as the free Hamiltonian with a random Cauchyconstant added to it and so does the Maryland model with the same λ (note that if θ is uniformly distributed on [0 , π ], then λ tan( θ ) is Cauchydistributed with half width λ ).In addition to a proof of completeness of the eigenfunctions foundby Prange’s group, [639] studies other properties of this model includ-ing the fact that for α a Liouville number the spectral measures are 18 B. SIMON purely singular continuous and the structure of the (non-normalizable)eigenfunctions for such α . I note that this model was one of the firsttimes that non-mathematical physicists had to face singular continuousspectrum; they gave the corresponding eigenfunctions which decay inan average sense but are not normalizable, the name “exotic states.”(4) Clock Spacing of Zeros for Ergodic Jacobi Matrices with Abso-lutely Continuous Spectrum We saw above that the DOS describes thebulk properties of the eigenvalue distribution of Schr¨odinger operatorsin large boxes in that the eigenvalue counting distribution convergesto the DOS. But there is the issue of the fine structure, in particularthe spacing between nearby eigenvalues. The earliest results on thisproblem are in the random Anderson type case where Molchanov [474]in 1 D and Minami [472] in higher dimensions proved the distributionis asymptotically Poisson. For random matrices, the fine spacing inthe bulk is governed by the Wigner surmise for GOE/GUE (see Mehta[467] or Deift [130] for discussion, proof and history). Basically, becauseof strong localization in the Anderson case, nearby eigenvalues don’timpact each other and the placement of such eigenvalues are close toindependent of each other. In the random matrix case, eigenfunctionsare more spread out, there is some eigenvalue repulsion so eigenvaluesare less likely to be too close to each other.I gave the problem of extending the Poisson results to OPUC tomy then graduate student, Mihai Stoiciu which he solved [699]. Alongthe way, I suggested that he do some numeric calculations and, forcomparison, suggested he look at some rapidly decaying Verblunskycoefficients. Here is the striking result that he found for the zeros ofΦ n =20 when α j = (cid:0) (cid:1) j +1 : WELVE TALES 119 Figure 3. Zeros of an OPUCBy a Theorem of Mhaskar-Saff [469], it was known that the countingmeasure for the zeros in this case converges to a uniform distribution onthe circle of radius but I was amazed when I saw that the eigenvaluerepulsion was so strong they seemed to spaced like the numerals on aclock. I called this “clock spacing”, a name which stuck even wheneventually applied to OPRL when the spacing was locally rigid butglobally not equally spaced because the limiting DOS didn’t have auniform density. I wrote a series (the last with Last) [667, 666, 668, 417]for situations there Verblunsky or Jacobi parameters converged to aconstant or periodic sequence.Motivated by this, Lubinsky [453, 452], found a new approach toclock behavior for OPRL with a.c. spectrum [ − , 1] based on provinga universality result for the Christoffel-Darboux kernel K n ( x, y ) = n (cid:88) j =0 p j ( x ) p j ( y ) (9.29) ? ? namely that, one says that bulk universality holds at x if and only if K ( x + a/n, x + b/n ) K n ( x , x ) → sin( πρ ( x )( b − a )) πρ ( x )( b − a )) (9.30) uniformly in bounded a, b where ρ ( x ) is the weight in the DOS which isassumed to be absolutely continuous. Earlier, although Lubinsky andI didn’t realize it until later, Freud [194] had studied the fine structureof zeros, had proven (9.30) in a less general setting and realized that itimplied clock spacing in the sense that if . . . x [ n ] − k < . . . < x [ n ] − < x [ n ]0 ≤ 20 B. SIMON x < x [ n ]1 < . . . are the zeros near x , thenlim n →∞ n ( x [ n ] j +1 − x [ n ] j ) = 1 /ρ ( x ) (9.31) ? ? for all j ∈ Z . Levin [430] independently rediscovered this connection,so the fact that bulk universality implies clock behavior is sometimescalled the Freud-Levin theorem.Lubinsky proved bulk universality for a large class of measure sup-ported on [ − , 1] with a.e. non-vanishing a.c. weight there. Totik [720]and I [671] were able to replace [ − , 1] by fairly general sets e ⊂ R but we required that e have a large interior, so large it was dense in e . Last and I realized that Lubinsky’s second approach [452] mightallow one to handle various almost periodic Jacobi matrices with a.c.spectrum even though that spectrum is nowhere dense (e.g. the AMOwith | λ | < 2) but we ran into a couple of hard technical problems. For-tunately, we were able to convince Avila to attack these issues and thethree of us [23] were able to prove bulk universality and clock behaviorfor a.e. x in σ ac for a large class of almost periodic Jacobi matrices.I wrote my OPUC books [664, 665] with one goal to extend thespectral analysis of Jacobi matrices to OPUC (i.e. replacing Jacobiparameters by Verblunsky coefficients). Included were two sectionsconcerning ergodic Verblunsky coefficients, one on random [665, Section12.6] and one on subshifts [665, Section 12.8], a class of weakly almostperiodic functions.I end this section on almost periodic Jacobi matrices and Schr¨odingeroperators by emphasizing that because it has focused on my own work,there is no discussion of some issues and limited discussion of later workon the issues we do discuss. In particular, I have not said anythingon Hausdorff dimension of the spectrum in the case where σ ( H ) hasLebesgue measure zero nor about long time behavior of powers of theposition (although I do have two papers on the latter [517, 644]). Norhave I discussed subshifts and substitution models except for the OPUCwork just mentioned and [298]. For more on these subjects, the readercan consult a number of books and review articles [2, 91, 119, 120, 337,339, 414, 499] (that said, we really do need a more recent comprehensivereview of the AMO).10. Topological Methods in Condensed Matter Physics (cid:104) s10 (cid:105) I was a pioneer [96] in the use of topology and geometry (mathemati-cians sometimes use “geometry” when there is an underlying distanceand “topology” for those geometric object that don’t rely on a dis-tance) in NRQM. In particular, Avron, Seiler and I [36] realized that WELVE TALES 121 the approach of Thouless et al. [717] to the quantum Hall effect (forwhich Thouless got the Nobel prize) was basically an expression of thehomotopic invariants (aka Chern integers) of a natural line bundle thatarises in certain eigenvalue perturbation situations, and I realized [632]that the phase that Berry [66] found in the quantum adiabatic theo-rem is holonomy in this bundle and that the quantity Berry [66] usedto compute this phase (and which independently had been found byAvron et al. [36]), now called the Berry curvature , is just the curvaturein this line bundle. I emphasize that Thouless et al. [717] never men-tion “topology” and that Thouless learned they’d found a topologicalinvariant, essentially the Chern class, from me. And the only mentionof curvature or holonomy in Berry [66] is where he remarks that BarrySimon, commenting on the original version of this paper, points outthat the geometrical phase factor has a mathematical interpretation interms of holonomy, with the phase two-form emerging naturally (in theform (7 b)) as the curvature (first Chern class) of a Hermitian linebundle .As a mathematician, I am mainly an analyst and most of my trainingand expertise is analytic so, as background, I should explain somethingabout how I came to know enough toplogy/geometry to realize its sig-nificance in NRQM. As a freshman at Harvard, I took the celebratedMath 55 Advanced Calculus course whose first half did differential cal-culus in Banach spaces and second half integral calculus on manifolds.This was a dip into the sea of geometry but from an analytic point ofview without any discussion of Riemannian metrics or curvature. I didsome self study of general relativity but the true topology/geometrywas hidden since my study was in physics books (and before the eraof those that emphasized the geometry). A key part of my educationwas a course on Algebraic Topology given my senior year by ValentinPo´enaru , then a recent refugee from Romania, who was visiting Har-vard. It was a wonderful course and I really got into the subject, somuch that Po´enaru took me aside and tried to convince me to give upmathematical physics and switch to topology. I was particularly takenwith the homotopy group long exact sequence of a fibration. (SeeHatcher [281] for background on this and other topological issues).Let me mention one of the simplest examples of fibrations of in-terest in physics, namely, the Hopf fibration, which is a naturalmap of S to S . Let σ j ; j = 1 , , σ matri-ces. If a = ( a , a , a , a ) = ( a , −→ a ) is a unit vector in R , then U ( a ) ≡ a + i −→ a · σ is a unitary matrix with determinant 1 if andonly if a ∈ S . In that case there is a rotation R ( a ) on S defined by 22 B. SIMON U ( a )( b · σ ) U ( a ) − = R ( a )( b ) · σ . This is the Cayley-Klein parametriza-tion of rotations, a map of SU (2) onto SO (3). If e is a unit vector inthe z direction, then a (cid:55)→ R ( a ) e defines the Hopf fibration, H , whichmaps S to S . The point is that it is easy to see (for example, bylooking at the inverse images of the north and south poles) that in-verse images of distinct points under H are circles which are linked sothe map is homotopically non-trivial proving that π ( S ) is non-zero(in fact, this homotopy group is generated by H and is just Z ).Of course, geometry in the naive sense was present, even central,to some of my work in 1970’s, for example the work on phase spacemethods in N-body NQRM (see Section 6) and I had even mentionedthat the Agmon metric was the geodesic distance in a suitable Rie-mann metric but if one thinks of “real” geometry needing curvatureand “real” topology needing homology or homotopy invariants, I’d notused them in my research in the ’70’s. In Section 8, I mention workthat was motivated by Witten’s seminal paper [738] on the supersym-metry proof of the Morse inequalities and index theorem. This paperhas been celebrated not only for the results itself but because of thebridge it opened up between high energy theorists studying gauge (andlater string) theories and topologists but it also impacted me in leadingme to consider certain geometric ideas that I needed in the work I’lldescribe in this Section. This is not so much in those of my papers di-rectly motivated by Witten [633, 637] but through other mathematicsmotivated by it. For Witten motivated several reworkings of the proofof the Atiyah-Singer Index theorem, in particular, a preprint of Getzler[231] (see [117, Chapter 12] for additional references) which caught myattention in the period just after I gave the Bayreuth lectures whicheventually appeared as [117]. I had lectured on Witten’s proof of theMorse inequalities there and decided to add a chapter on this furtherextension (the chapter, chapter 12, was actually the only chapter Iwrote in [117] - the other chapters were written by my coauthors basedon and usually expanding the lectures I’d given).For pedagogical reasons, I decided to give details only in the special,indeed, classical case of the Gauss-Bonnet theorem where it turns outthat Getzler’s proof is essentially one found in 1971 by Patodi [501] whodidn’t know that he was speaking supersymmetry! While I’d heard ofthe Gauss-Bonnet theorem, I hadn’t known exactly what it said untilfollowing up on Witten taught me all about it. Since it will explainsome of my later work, let me say a little about this theorem (and alsoholonomy) in the case of S , the sphere of radius R embedded in R .At each point, the Gaussian curvature is 1 /R so, if K is the curvature WELVE TALES 123 and d ω the surface area, we have that12 π (cid:90) K dω = 12 π R πR = 2 (10.1) The remarkable fact is that if you deform the sphere to another sur-face, say, an ellipsoid, then the curvature is no longer constant but theintegral in (10.1) is still 2. But this is not true for the torus. Theintegral is still independent of the underlying metric needed to define K , but it is 0, as can been seen by looking at the flat torus R / Z withthe Euclidean metric on R (which cannot be isometrically embeddedinto R but can in R ). In fact, for any surface in R (and for hyper-surfaces in general dimension) the integral is the Euler characteristic ofthe surface (Euler-Poincar´e characteristic in higher dimension). This isthe Gauss-Bonnet theorem. It says that the integral of a natural geo-metric quantity lies in a discrete set and is determined by topologicalinvariants.To explain holonomy, consider someone carrying a spear around theearth trying at all times to keep the spear tangent to the sphere andparallel to the direction it was pointing (which may or may not beparallel to the direction the person is walking). Imagine, going alongthe equator through one quarter of the earth, turning left, going to thenorth pole, turning left and going back to the original point. Supposethe spear is parallel to the equator at the start. The person turns tomove along a line of longitude, but being careful not to turn the spear,it will point directly to his right. After the next turn, the spear willpoint backwards. So despite having tried to keep it parallel , uponreturn, it has rotated by 90 ◦ , i.e. π/ holonomy . The path encloses one eighth of theearth, a area of 4 πR / πR / 24 B. SIMON trademark New York accent: “Whadya mean? He’s a Professor, ofcourse he knows it!” I might have recalled all that when I neededprecisely homotopy and the exact sequence of a fibration several monthslater in my work with Avron and Seiler, but it helped that I’d had thisinteraction.In early 1983, Yosi Avron told me about the paper of Thouless et al.paper [717] which gave a novel explanation of the quantum Hall effect,a subject that had fascinated Yosi. The striking aspect of that effectis that a resistance was quantized. In the TKNN approach (we quicklycame up with that abbreviation, sometimes TKN , especially TKNNintegers, a name which has stuck), this arose because, using the Kuboformula, they got the resistance (in a certain idealized situation) wasgiven by an integral over a torus that turned out to be an integer (insuitable units).We quickly realized that their integers were associated to a singleband which was assumed non-degenerate (i.e. at every point in theBrillouin zone, the eigenstate for that band is simple) and their inte-grand involved the change of eigenfunction. We also realized that sincethe integrand was an integer it had to be invariant under continuouschange and so an indication of a homotopy invariant of maps from thetwo dimension torus T to unit vectors in Hilbert space mod phases(equivalently a continuous assignment of a one dimensional subspace inthe Hilbert space to each point in T ). After more thought and study,we learned that the homotopy class of maps from T could be classifiedby maps from S and S and so the underlying homotopy groups of P ( ∞ ), the one dimensional subspaces of a Hilbert space. We also con-sidered that there might be non-trivial homotopy invariants dependingon several bands so what we wanted to consider was the homotopygroups of the set, N , of compact operators with non-degenerate eigen-values. We got excited since if, for example, we found a non-trivial π ,there would be new topological invariants for the physically relevantthree-dimensional torus.By a continuous deformation, we could consider maps to a fixedset of simple eigenvalues but variable eigenspaces. Given the phasechange this was the same as the quotient of all unitary maps by thediagonal unitary maps U ( H ) /D U ( H ). So these homotopy groups mightbe computable via the exact sequence of the fibration that my talk withFeynman had reminded me about! Indeed, since it was known that theset of all unitaries U ( H ) is contractible, it has no homotopy, i.e. allhomotopy groups are trivial and, thus, by the exact sequence of thefibration, we knew that π j ( N ) = π j − ( D U ( H )). Since the diagonalunitaries is just an infinite product of circles, T , and π j ( T ) is Z for WELVE TALES 125 j = 1 and 0 for all other j , we had discovered that the only non-trivialhomotopy group of N was π , that the same was true for P ( ∞ ) andthat π ( N ) was just an infinite product of π ( P ( ∞ ))’s. In other words,the only homotopy invariants were the TKNN integers.We added Reudi Seiler, whom Yosi had been consulting, to the au-thors and published this negative result in Physical Review Letters [36].We made a big deal of our new result that if two non-degenerate bandswith TKNN integers n and n went through a degeneracy as param-eters were varied so that afterwards they were again non-degeneratewith TKNN integers n and n , then n + n = n + n . But therewere results that were more important although only noted in passing.Most basic was the new one that the TKNN integers were homotopyinvariants, something that would be clarified by my work on Berry’sphase which I turn to shortly. We also found two compact formulae forthe integrand that eventually became commonly used in further work.First if that ψ j is the eigenstate of band j , then the correspondingTKNN integer, n j , is given by n j = 12 π (cid:90) T K j ; K j = i (cid:104) dψ j , dψ j (cid:105) (10.2) We were especially fond of a second formula, that if P j is the projectiononto ψ j , then K j = i Tr( dP j P j dP j ) (10.3) We liked this because while (10.2) requires a choice of phase in eachspace, (10.3) is manifestly phase invariant. The operator d in the lasttwo expressions is the exterior derivative and there is an implicit wedgeproduct. The reader might worry that because df ∧ df = 0, if therewere no trace and P j in (10.3) was a function, the quantity would be0. But because P j is operator valued, it is not 0. Indeed, K = i (cid:88) k,(cid:96) Tr (cid:18) ∂P j ∂x k P j ∂P j ∂x (cid:96) (cid:19) dx k ∧ dx (cid:96) = i (cid:88) k,(cid:96) Tr (cid:18) P j (cid:20) ∂P j ∂x k , ∂P j ∂x (cid:96) (cid:21)(cid:19) dx k ∧ dx (cid:96) = i (cid:88) k,(cid:96) (cid:28) ψ j , (cid:20) ∂P j ∂x k , ∂P j ∂x (cid:96) (cid:21) ψ j (cid:29) dx k ∧ dx (cid:96) (10.4) where [ · , · ] is commutator and we used the antisymmetry of dx k ∧ dx (cid:96) .The next part of this story took place in Australia, so I should men-tion that trip in the summer of 1983 (well, the winter in Australia!)almost didn’t happen. My fourth child, Aryeh, was born in Dec., 1982 26 B. SIMON and given the time to get his birth certificate and passport, it was onlythe end of April that I was able to contact the Australian consul in LosAngeles to get visas for all of us including a work visa for me. He senta long medical form for me requiring a new general exam from a doctorand xray. I’d had them 3 months before at Kaiser but was told by theconsul that I had to do them over. I’ve been raised to avoid unneces-sary xrays and I wasn’t sure Kaiser would agree to a second exam. Asfar as I could tell, this was a restriction put in place to make it difficultfor Asians to come and work and I tried to use my invitation fromthe Australian Academy of Sciences to get a waiver. The consul wasuncooperative, almost nasty. This was not only pre-Skype but emailwas almost non-existent and intercontinental phone calls were very ex-pensive, so I sent a telex to my host, Derek Robinson, explaining that,because of visa issues, I would probably have to cancel my trip. Thenext day, he called me, which impressed me given the cost of interna-tional calls, telling me to stay calm and he’d fix it. I didn’t know thatDerek was the secretary of the Australian Academy of Sciences. Butthree days later, I get a call from the consul saying “Sir, I am anxiousto issue your visas, but I need you to return the forms I sent you.” Ireplied “But what about the medical form.” “Oh, you don’t need that,sir.” According to the current vogue, I should feel guilty for havingused my white privilege, but given how important this visit turned outto be, I am glad.Derek was actually away for the first two weeks of my visit but BrianDavies had also just arrived so we collaborated together on the workon ultracontractivity that is mentioned in Section 3. Midway throughmy visit, I heard that Michael Berry, whom I’d meet several yearsbefore at Joel Lebowitz’ seminar, was visiting physics at AustralianNational University where Derek was in mathematics and where I wasvisiting. He’d given a seminar, but before I’d learned he was there,so I called and asked him for a private version which he kindly agreedto. He explained to me his work on an extra phase he’d found in theadiabatic theorem (see below) and gave me a copy of the manuscript[66] that he’d recently submitted to Proc. Roy. Soc. He mentioned thatBernard Souillard, when he heard about Berry’s work, told Berry thathe thought it might have something to do with the paper of Thoulesson TKNN integers but then Berry added that when he asked Thoulessabout it, Thouless said that he doubted there was any connection (nopun intended). I replied I thought there probably was and that night,I figured out all the main points that appeared in my paper [632]!Berry’s paper dealt with the quantum adiabatic theorem. This the-orem deals with a time dependent Hamiltonian H ( s ); 0 ≤ s ≤ WELVE TALES 127 considers T large and H ( s/T ) so one is looking at very slow changes. ϕ T ( s ) ≡ (cid:101) U T ( s ) ϕ ; 0 ≤ s ≤ T solves ˙ ϕ T ( s ) = − iH ( s/T ) ϕ T ( s ); ϕ T (0) = ϕ . Let E ( s ) be an isolated, simple eigenvalue of H ( s ) and let P ( s ) bethe projection onto the corresponding eigenspace. The adiabatic the-orem says that if P (0) ϕ = ϕ , then lim T →∞ ( − P ( s/T )) ϕ T ( s/T ) = 0,i.e. if you start in an eigenspace you stay in it adiabatically. Berryasked and answered the question, what happens if H (1) = H (0) soyou end where you start. What is the limiting phase of ϕ T ( T ). Thesurprise he found (it turned out that in 1956 Pancharatnam [495] haddone the same thing, but it had been forgotten) is that the naive guessthat ϕ T ( T ) ∼ e − iT (cid:82) E ( s ) ds ϕ is wrong but that there is an additionalphase, e i Γ . In my paper, I gave Γ the name it is now known by - Berry’sphase .Berry originally wrote Γ as a line integral but, then, assuming thatthe family H ( s ) was a closed curve in a parameter space, he used Stokestheorem to write Γ as the integral over a surface, S , in parameter spacewhose boundary was the closed curve in the formΓ = (cid:90) S K ( ω ) dω (10.5) ? ? K = Im (cid:88) m (cid:54) =0 (cid:104) ϕ m ( ω ) , ∇ H ( ω ) ϕ ( ω ) (cid:105) × (cid:104) ϕ ( ω ) , ∇ H ( ω ) ϕ m ( ω ) (cid:105) ( E m ( ω ) − E ( ω )) (10.6) where he supposed the interpolating Hamiltonian H ( ω ) had a completeset { ϕ m } m of simple eigenfunctions with H ( ω ) ϕ m ( ω ) = E m ( ω ) ϕ m ( ω )and P ( ω ) ϕ ( ω ) = ϕ ( ω ); E ( ω ) = E ( ω ).What I did in my paper [632] is realize that what Berry was doingwas simple and standard geometry in the exact same setting as TKNN.I’d learned in the meantime that the TKNN integers were called theChern invariant and the curvature K was called the Chern class andused those names for the first time in this context. The adiabatictheorem defines a connection, i.e. a way of doing parallel transport andBerry’s phase was nothing but the holonomy in this connection. Berryhad used (10.2) as an intermediate formula in his paper but didn’thave the phase invariant formula of Avron-Seiler-Simon. Despite thefact that our independent work was earlier (dates of submission for ourpaper is May 31, 1983 and his June 13, 1983) and that the geometricideas were in our paper (and more explicitly with the name curvaturein [632]), K is universally known as the Berry curvature .Berry also realized that in situations where the parameter spacecould be interpolated into higher dimensions, that eigenvalue degen-eracies were sources of curvature, a theme I developed in [632]. 28 B. SIMON In the vast literature related to these issues, I should mention twoespecially illuminating points. The first involves the fact that the firstmathematically precise and, in many ways, still the best proof of thequantum adiabatic theorem is Kato’s 1950 proof [351] (see [682, Section17] for an exposition). Without loss, one can suppose E(s)=0 (other-wise replace H ( s ) by H ( s ) − E ( s ) ). Kato constructs a comparisondynamics solving dds W ( s ) = iA ( s ) W ( s ) , ≤ s ≤ W (0) = (10.7) ? ? iA ( s ) ≡ [ P (cid:48) ( s ) , P ( s )] (10.8) for which W ( s ) − P ( s ) W ( s ) = P (0) (10.9) ? ? by an explicit calculation and he proves that (cid:107) W ( s ) P (0) − U T ( s ) P (0) (cid:107) = O(1 /T ) (10.10) ? ? The relevant point here is that W ( s ) defines a connection whose dif-ferential, by (10.8), is [ P, dP ] so that its differential, the curvature, isgiven by (10.4). Thus the Avron-Simon-Seiler formula for the Berrycurvature was almost in Kato’s paper nearly 35 years before!Secondly, as noted in [632], when the Hilbert space is C n , this con-nection appeared a 1965 paper of Bott-Chern [74]. As noted laterby Aharonov-Anadan [6], this connection is induced by a Riemannianmetric going back to Fubini [206] and Study [702] at the start of thetwentieth century.I returned to the subject of the quantum Hall effect and Berry’sphase twice. As background, I note that from Berry’s paper onwards,a key observation was that Berry’s phase is zero if all the H ( s ) canbe taken simultaneously real (indeed, Berry tells the story that priorto this work, he noted a curiosity in eigenvalue perturbation theory; ifone has real matrices depending on two parameters with an eigenvaluedegeneracy only at 0 ∈ R , then going around the degeneracy causes asign flip in the eigenvector. In this case, because eigenvectors are chosenreal, there is only a ± degeneracy and so a unique way of continuing. Hetalked about this result and someone asked him what happened in thecomplex case and he replied, there was no difference. But after the talk,he realized that in the complex case, phase ambiguity meant there wasno unique way to continue under just perturbation of parameters andthen, that the adiabatic theorem did give a way of continuing which inthe complex case could lead to a non-trivial phase). Since the curvaturemust be real, the i in (10.4) (or the Im in (10.6)) show if all the P ’sare real then K = 0 and there is no Berry phase. For spinless particles, WELVE TALES 129 time reversal just complex conjugates the wave function so the mantrabecame “time reversal invariance kills Berry’s phase”. Magnetic fieldsdestroy reality of the operators (and are not time reversal invariant).Indeed, the basic example is to take a constant magnetic field, B ∈ R and H ( B ) = B · σ where σ is a spin s spin. The curvature is then(2 s + 1) B /B .In work with Avron and two then postdocs Sadun and Seigert[34, 35], I discovered that for fermions you could have a non-zero Berryphase even with time reversal invariance and that there was a remark-able underlying quaternionic structure relevant to their study. Theunderlying issue goes back to a 1932 paper of Wigner [735] on time re-versal invariance, T , in quantum mechanics. He first proved his famoustheorem that symmetries in quantum mechanics are given by eitherunitary or anti-unitary operators and then argued that T was alwaysantiunitary with T = for bosons and T = − for fermions. In theBose case, that means T acts like a complex conjugate and so the argu-ment of no Berry’s phase applies but not in the fermion case. Instead J ≡ T and, I , the map of multiplication by i are two anticommutingoperators whose squares are each − , so they and K = IJ turn the un-derlying vector space into one over the quaternions! It was Dyson [164]who first realized that fermions under time reversal have a quaternionicstructure (although he first notes the relevance of sympletic groups andthat the connection between such groups and quaternions is well knownin the mathematical literature on group representations).We worked out the details, especially for half integral spin systems.Just as the simplest example of Berry’s phase is a spin 1 / / × Yes, but some parts are reasonably concrete . While I had intro-duced topological ideas, I was somewhat dismayed about all theterribly fancy stuff that appeared in the math physics literature,especially throwing around the term “fiber bundle”. Yosi and Iused to joke that some people seemed to suffer from bundle fibro-sis. So we were concerned about some of the abstruse language in 30 B. SIMON Abstract Snark [518] that declared our abstract and oneother “almost Zen in their simplicity and perfection”.My other work in this area is three related papers that I wrote withAvron and Seiler [37, 38, 39] that followed up on an alternate approachto the quantum Hall effect due to Bellisard [55] in which topologyentered as an index in C ∗ -algebraic K-theory. We developed an indextheory for the simpler case where certain subsidiary operators wereFredholm. To me, some of the mathematics we developed was mostfascinating. In particular we proved ? (cid:104) T10.1 (cid:105) ? Theorem 10.1. Let P and Q be two orthogonal projections so that P − Q is trace class. Then Tr( P − Q ) is an integer. Remarks. 1. For discussion of trace class, see [680, Section 3.6].2. This is a result that begs to be proven by Goldberger’s method .3. Our proof relied on two operators used extensively by Kato [363], A = P − Q and B = 1 − P − Q which he showed obeyed A + B = . Wenoted [39] that one also had the supersymmetry relation AB + BA = 0.Since A is trace class and self-adjoint, using a basis of eigenfunctionsand the Hilbert-Schmidt Theorem [680, Theorem 3.2.1] shows thatTr( A ) = (cid:88) λ λd λ (10.11) where we sum over eigenvalues and d λ = dim( H λ ) with H λ = { ϕ | Aϕ = λϕ } . The supersymmetry implies that ψ ∈ H λ ⇒ Bψ ∈ H − λ .Since B (cid:22) H λ = (1 − λ ) , we see if λ (cid:54) = ± 1, then B is a bijectionof H λ and H − λ so, for such λ , we have that d λ = d − λ . Thus (10.11)implies that Tr( A ) = d − d − ∈ Z .4. Slightly earlier, this result was proven by different methods byEffros [169]. His proof is sketched in [680, Problem 3.15.20] and ourproof can also be found [680, Example 3.15.19]. I found another proofusing the Krein spectral shift which is sketched in [680, Problem 5.9.1].Amrein-Sinha [15] has a fourth proof. Murph Goldberger was one of my professors at Princeton (see the remark after(8.24)) and, in his day, a famous theoretical physicist who used to joke about thingsthat just had to be true: oh, you just use Goldberger’s method which is a proof byreductio ad absurdum. Suppose it’s false; why that’s absurd! WELVE TALES 131 5. For a review of some of the literature on pairs of projections,see [682, Section 5]. I have several more recent papers on pairs ofprojections [681, 75].11. Anderson Localization: The Simon-Wolff Criterion (cid:104) s11 (cid:105) I have some contributions to random Schr¨odinger operators, espe-cially in one dimension. While the first of my papers in the area pre-dates slightly the work of the last section, I’ve placed this here becausemy two most significant contributions were finished near the end of1984, so close to each other that there was a joint announcement [689]!One is the work with Tom Wolff [690] on a necessary and sufficient con-dition for point spectrum which appears in the title of this section andthe other is work with Michael Taylor [688] on regularity of the densityof states in the Anderson model. While the Simon-Taylor work wasdone first (indeed, I talked about it at the conference where I learnedof Kotani’s work that motivated Simon-Wolff), I begin with [690].The generalized Anderson model is described in Section 9. Supposethat the single site distribution, dκ , is acwrt Lebesgue measure. If H ω has dense point spectrum for a.e. ω , then, by independence at distinctsites, if we fix all sites but one, we will have dense point spectrum forLebesgue a.e. choice of the potential at the remaining point in thea.c. support of the single site distribution. So it is natural to discuss afamily of rank one perturbations, A λ = A + λQ, Q = (cid:104) ϕ, ·(cid:105) ϕ (11.1) where is A is a self-adjoint operator with simple spectrum (I discussin remark 1 why assuming simplicity is no loss), Q the projection ontoa unit vector ϕ and λ ∈ R . If dµ λ is the spectral measure for A λ invector, ϕ , a key role is played by the function K ( E ) = (cid:90) dµ ( E (cid:48) )( E − E (cid:48) ) (11.2) which is well-defined as a function with values in (0 , ∞ ] including thepossible value of ∞ . This function will play a crucial role in the nextsection also. One main theorem of Simon-Wolff [690] is (cid:104) T11.1 (cid:105) Theorem 11.1 (Simon-Wolff criterion [690]) . Suppose A is a self ad-joint operator with cyclic unit vector ϕ . Fix an open interval ( α, β ) in spec( A ) . The following are equivalent:(a) For (Lebesgue) a.e. real λ , A λ has dense point spectrum in ( α, β ) (b) For (Lebesgue) a.e. real E ∈ ( α, β ) , we have that K ( E ) < ∞ . 32 B. SIMON Remarks. 1. We supposed that A has simple spectrum, with ϕ acyclic vector. For general A and ϕ , we can restrict A to the cyclicsubspace generated by ϕ and that restriction obeys the simplicity andcyclicity assumptions, so we can conclude something about the spectralmeasure dµ λ . For an Anderson type model, if we know each δ β , β ∈ Z ν has a spectral measure dense pure point spectrum, we get the resultfor the full operator. We also note that it was later shown that anysingle δ β is cyclic in the localization region; see the end of this section.2. Simon-Wolff [690] further note that if G ( β, γ ; z ) = (cid:104) δ β , ( H − z ) − δ γ (cid:105) is the Green’s function and dµ is the spectral measurefor H and δ , then (cid:90) | E − E (cid:48) | − dµ ( E (cid:48) ) = lim ε ↓ (cid:88) β ∈ Z ν | G ( β, E + i(cid:15) ) | (11.3) The two main approaches to the spectral analysis of multidimen-sional generalized Anderson models are the multiscale analysis ofFr¨ohlich-Spencer (see [700]) and the method of fractional moments ofAizenmann-Molchanov (see [13]). Both most directly prove exponen-tial decay of Green’s functions with some kind of uniformity as oneapproaches the real axis and prove the finiteness of the right side of(11.3) for a.e. ω and a.e. E in some interval so, by Theorem 11.1, theyimply dense point spectrum.The proof of Theorem 11.1 relies on two elements - a general analysisof the spectral type under rank one perturbations due to Aronszajn [18]and Donoghue [159] (Aronszajn discussed the special case of variationof boundary condition for ODEs and Donoghue extended to generalrank one perturbations). We need the Stieltjes (aka Borel, aka Cauchy)transforms F λ ( z ) = (cid:90) dµ λ ( E (cid:48) ) E (cid:48) − z (11.4) ? ? and we define various subsets of R using F λ =0 and K : S λ = { x | lim ε ↓ F ( x + iε ) = − λ − } for λ (cid:54) = 0; S = { x | lim ε ↓ Im F ( x + iε ) = ∞} (11.5) ? ? P = { x | K ( x ) < ∞} ; P λ = S λ ∩ P for λ (cid:54) = 0; P = { x | lim sup ε ↓ ε Im F ( x + iε ) > } (11.6) WELVE TALES 133 L = { x | lim ε ↓ Im F ( x + iε ) ∈ (0 , ∞ ) } ; B = R \ (cid:32) (cid:91) λ ∈ R S λ ∪ L (cid:33) (11.7) ? ? where, when we write a lim is equal to some value, it includes thestatement that the limit exists.As preliminaries, we note first that, by the dominated convergencetheorem, if K ( x ) < ∞ , we have that lim ε ↓ F ( x + iε ) exists and liesin R so P is (cid:83) { λ (cid:54) =0 } P λ plus the set where the limit is 0. Secondly, thegeneral theory of Stieltjes transforms implies that each S λ has measurezero. Note also that the sets P λ are disjoint from each other and from L . We say that Z ⊂ R a supports of a measure , ν , if and only if ν ( R \ Z ) = 0 (11.8) ? ? Then the work of Aronszajn-Donoghue implies (cid:104) T11.2 (cid:105) Theorem 11.2 (Aronszajn-Donoghue [18, 159]) . Let A λ be a familyof rank one perturbations. Then(a) The a.c. parts of the measures dµ λ,ac are mutually absolutelycontinuous for all λ ∈ R and are supported on L .(b) The singular parts of the measures dµ λ,sing are mutually singularand for distinct λ ∈ R and each is supported on S λ .(c) For all λ ∈ R , the pure point part of the measure, dµ λ,pp issupported on P λ and the singular continuous part of the measure issupported on S λ \ P λ .(d) The set B has Lebesgue measure zero and, for all λ ∈ R , we havethat µ λ ( B ) = 0 . Remarks. 1. For proofs, see [680, Section 5.8] or [662, Section 12.2].2. One can say much more about P λ and L . First, L is the essentialsupport of the all the dµ λ,a.c. . Secondly, for λ (cid:54) = 0, each point, x in P λ is a pure point with dµ λ,pp ( x ) = ( λ K ( x )) − .3. After my introduction to rank one theory in the course of thiswork, I was motivated to do a lot more in the subject. First, the workon Baire generic singular continuous components [141, 139] discussedin Section 12. Secondly, I worked on the natural meaning of A λ when λ = ∞ [218] and the extension of the theory when A is unbounded and ϕ is very singular [384]. Finally, I extended the theory to multiplicativerank one perturbations of unitary operators, a subject useful in OPUC[664, 669].4. After those works, I wrote some lecture notes on rank one pertur-bations [650]. When the AMS decided to reprint my trace ideals book, 34 B. SIMON which had gone out of print, it made sense to include those notes assome extra chapters in the second edition [662].The second element of the Simon-Wolff analysis was our result thathas come to be called spectral averaging : (cid:104) T11.3 (cid:105) Theorem 11.3 (Spectral Averaging [690]) . For general rank one per-turbations, one has that (cid:90) [ dµ λ ( x )] dλ = dx (11.9) in the sense that (cid:90) [ f ( x ) dµ λ ( x )] dλ = (cid:90) f ( x ) dx (11.10) ? ? for any continuous function, f , of compact support on R . Remarks. 1. Theorems 11.2 and 11.3 immediately imply Theorem11.1 because, by Theorem 11.2, (a) is equivalent to dµ λ (( α, β ) \ P ) = 0for a.e. λ and by Theorem 11.3, (cid:82) [ dµ λ (( α, β ) \ P )] = | ( α, β ) \ P | .2. There are variants of spectral averaging that predate [690]. In1971, Javrjan [332] proved equivalent formulae for the special case ofboundary condition variation of Sturm-Liouville operators on [0 , ∞ ).For some applications all that is needed is the consequence of spectralaveraging that if a set Q ⊂ R has Lebesgue measure zero, then for a.e. λ one has that dµ λ ( Q ) = 0 for a.e. λ . This fact (or the stronger onethat some average of dµ λ with an a.e. positive weight is dominatedby an a.c. measure) appears in the literature in several place prior to[690]: for example Carmona [89] and Kunz-Souillard [402].I conclude the discussion of Simon-Wolff [690] with a bit about thehistory of its genesis. In the summer of 1984, Kotani reported on someinteresting work at a conference in Maine. I didn’t hear about thiswork until he and I attend a conference in Bremen in Nov althoughhe eventually published his work in the Proceeding of the conferencein Maine [394]. While Kotani focused, as he often did, on continuumSchr¨odinger operators, I’ll discuss the discrete case which he mentionedin passing. He looked at an ergodic discrete Schr¨odinger operators ona half line ( n ≥ 1) (i.e. a n ≡ b n ( ω ) samples of an ergodic process) inan energy region, ( α, β ) where one knew the Lyaponov exponent waspositive. He considered operators h θω where the eigenfunction equation hu ( n ) = u ( n + 1) + u ( n − 1) + b n u ( n ) = Eu ( n ) , n ≥ θ ) u (1) + sin( θ ) u (0) = 0. Thisis equivalent to truncating the doubly infinite matrix but replacing b WELVE TALES 135 by b − cot( θ ). As explained in Section 9, one has exponentially grow-ing or decaying solutions except for an ω -dependent set of energies in( α, β ). By making explicit an argument of Carmona [89], he showedthat for Lebesgue a.e. θ , the spectral measures were supported on theset where one had this exponential dichotomy. Thus he had the shock-ing result that in cases like the one where Avron and I proved therewas purely singular spectrum (AMO with large coupling and Liouvillefrequencies), the half line problem had pure point spectrum for a.e.boundary condition θ .It was immediately clear to me that these ideas might say somethingabout dense point spectrum for the higher dimensional Anderson modelwhere Fr¨ohlich-Spencer had recently announced results on exponentialdecay of Green’s functions. I asked Kotani if he’d thought about suchapplications and when he said no, I asked if he minded if I thoughtabout it and he said fine. I returned to Caltech and quickly realizedthe relevance of the Aronszajn-Donoghue theory and understood thekey was finding some abstract version of the Carmona argument. Idecided it was a question connected with Hilbert transforms and soconsulted Wolff and we came up with spectral averaging.Bernard Souillard was also at the conference in Bremen and he alsorealized the possible applicability of Kotani’s scheme to multidimen-sional localization and he, together with Delyon and L´evy also devel-oped an approach [144, 145, 146] to these problems. They did notphrase it in terms in general rank one perturbations and required ex-ponential decay (rather than only (cid:96) decay) and didn’t have a necessaryand sufficient theorem so, Simon-Wolff has been much more generallyquoted. But their ideas worked more easily in some non-rank one situ-ations and, indeed, Delyon, Souillard and I [149] used their approach toprove some results about random operators with so-called off-diagonaldisorder (which are rank 2)!After my work with Wolff, I wrote two papers, one with Kotani[642, 397] applying these ideas to discrete Schr¨odinger operators instrips.Before leaving the subject of spectral averaging, I should mentiona later work of mine [656] that extends it to trace class perturbationsand averages over finite intervals and relates it to a wonderful formulaof Birman-Solomjak [71]. It involves the Krein spectral shift (see [662,Section 11.4] for references and the theory), ξ A,B ( x ) which, whenever B − A is trace class, can be defined byTr( f ( B ) − f ( A )) = (cid:90) f (cid:48) ( x ) ξ A,B ( x ) dx (11.11) ? ? 36 B. SIMON Javrjan [332] actually had a local version of (11.9) which generalizedto arbitrary rank one perturbations says that (cid:90) λ λ [ dµ λ ( x )] dλ = ξ A + λ Q,A + λ Q ( x ) dx (11.12) ? ? from which (11.9) follows because lim λ →∞ ξ A − λC,A + λC ( x ) = rank( C ) (for all x ) if C is finite rank. The main result in [656] considers generalfamilies, A ( s ); s ≤ s ≤ s , of self-adjoint operators with a weak deriv-ative C ( s ) which is trace class, positive and continuous in trace norm.I defined dµ s ( x ) = Tr( C ( s ) / dE s ( x ) C ( s ) / ) (with A ( s ) = (cid:82) xdE s ( x )in the spectral resolution form of the spectral theorem [680, Section5.1]) and proved that (cid:90) s s [ dµ s ( x )] ds = ξ A ( s ) ,A ( s ) ( x ) dx (11.13) ? ? I showed that this was equivalent to the formula of Birman-Solomjakthat dds Tr( f ( A ( s ))) = T r ( C ( s ) f (cid:48) ( A ( s ))) (11.14) and provided a half page proof of (11.14).Turning next to my work with Taylor [688] which concerns the issueof regularity of the IDS, k ( E ). The most heavily quoted and usedregularity result is the estimate of Wegner [726] that for the generalizedAnderson model on Z ν , if the b n are iidrv with distribution (9.2) where dκ ( x ) = g ( x ) dx (11.15) with g ∈ L ∞ , then one has the Wegner estimate | k ( E ) − k ( E (cid:48) ) | ≤ (cid:107) g (cid:107) ∞ | E − E (cid:48) | (11.16) ? ? This estimate is easy to prove and can be deduced from spectral aver-aging (although it predates it!). It (or rather its finite volume analog)is the starting point for most variants of multiscale analysis. This esti-mate and others that are known in general dimension are of the formthat k is as regular as E (cid:55)→ κ ( −∞ , E ). What Taylor and I proved wasthe possibility of significantly greater smoothness in one dimension (atleast once κ has some minimal smoothness).Why did I think this might be true? For the free case, k ( E ) = , if E ≤ − π arccos (cid:0) − E (cid:1) , if − ≤ E ≤ , if E ≥ ? ? which implies that k ( E ) is C ∞ on ( − , 2) with dk/dE = (cid:2) π √ − E (cid:3) − so there is a singularity in dk/dE at E = ± 2. In WELVE TALES 137 general, one would expect there are singularities at the edges of thespectrum. Indeed, this k ( E ) is globally H¨older continuous of order 1 / k ( E ) went to zero as E ↓ Σ − ,the bottom of the spectrum faster than the inverse of any power of( E − Σ − ) − consistent with k ( E ) being C ∞ as E shifts from the spec-trum to below the spectrum. I discussed this with Tom Spencer whowas dubious that k was C ∞ for the original Anderson model, so wemade a 25 cent bet on whether it was true (I would win if someone,not necessarily me, proved it true and he would win if someone, notnecessarily him, proved it false).Of course one expected this not merely for the Anderson model(where g is the characteristic function of an interval), but for at leastsome generalized Anderson models. I found that one needed some min-imal regularity on κ because Bert Halperin [278] had shown that therewere examples where dκ was a two point measure, where k was noteven C . While Halperin went on to become a distinguished condensedmatter theorist, he wrote this paper as a junior undergrad at Harvard!While the argument was solid, there were missing points of mathemat-ical clarity, so that Taylor and I, who wanted to advertise the result,included details in an appendix [688]. The model has dκ ( x ) = (1 − θ ) δ ( x ) + θδ ( x − λ ) (11.18) ? ? with 0 < θ < / 2. This model came to be called the Bernoulli-Anderson model . We showed that k ( E ) was not H¨older continuousof any order larger than α = 2 | log(1 − θ ) | / Arc cosh(1 + | λ | ) so bytaking θ small and/or λ large, one can assure lack of H¨older continuousof any prescribed order. We also gave heuristics and conjectured thatfor those extreme values dk should have a singular component (we recallthat the Cantor function is H¨older continuous of order log(2) / log(3)so dk can be singular continuous even those k is H¨older continuous).Motivated by this Carmona et. al. [90] proved this conjecture (andmore importantly proved localization in Bernoulli Anderson models)and Martinelli-Micheli [461] even proved for any fixed θ , dk was purelysingulary continuous for all large λ .The main result of Simon-Taylor is (cid:104) T11.4 (cid:105) Theorem 11.4. Let k be the IDS for a generalized Anderson modelin ν = 1 dimension with dκ of the form (11.15) where g has compactsupport and for some α > , one has that (1 + k ) α/ ˆ g ( k ) is the Fouriertransform of an L function. Then k is C ∞ . 38 B. SIMON When g is the characteristic function of an interval, the hypothesisholds for any α < 1, so this won my 25 cent bet with Spencer! Letme say something about the strategy and genesis of this result. Mostof the early proofs of localization in the 1D Anderson model reliedon a theorem of Furstenberg [208], who proved that, under certain cir-cumstances, products of iidrv SL (2 , R ) matrices had positive Lyaponovexponent. His proof relied on the action of SL (2 , R ) on RP (1), real pro-jective space (by ( A, [ ϕ ]) (cid:55)→ A [ ϕ ]) and the induced natural convolutionof measures on SL (2 , R ) with measures on RP (1) to get measures on RP (1). If µ was the probability measure on SL (2 , R ) describing thedistribution of individual matrices in the random product, Furstenbergshowed and used that there was a unique measure ν on RP (1) so that µ ∗ ν = ν . In the Anderson case, for each real energy, E , there isa distribution of transfer matrix (9.11) and so an invariant measure, ν E for each E . I realized that by a discrete analog of the Sturm os-cillation theorem, k ( E ) was the weight that ν E gave to those lines in RP (1) with two coordinates of the same sign so that smoothness of k should be implied by smoothness of ν E in E . Since ν E was also invari-ant for multiple SL (2 , R ) convolutions of µ E , what one needed is thatthese multiple convolutions got smoother and smoother in E . While Iwas interested in smoothness in E , I suspected (correctly it turns out)that what one really needed was that these high order convolutions of µ E were a.c. wrt Haar measure on SL (2 , R ) with weights that weresmoother and smoother in the group parameters.This was a question in noncommutative harmonic analysis and I as-sumed the representation theory of SL (2 , R ) would play a major role, soI contacted Michael Taylor, who I’d heard was a big expert on the topic(shortly after this, he published two books on the subject [711, 712])and suggested that we work on it. At some point, I also spoke toEli Stein who was also a big expert on the representation theory of SL (2 , R ) and he made the suggestion that it is often easier to controlconvolutions on SL (2 , R ) with one’s “bare hands” rather than by usingthe non-commutative Fourier transform which is what we did. Theunderlying µ E are certainly not a.c. wrt Haar measure on SL (2 , R )since they are supported on a one dimensional subset of the three di-mensional group but we proved that under the technical condition on g in Theorem 11.4 the three fold convolution is a.c. wrt Haar measurewith a weight that has a tiny bit of smoothness so that in the stan-dard way, the higher order convolutions of that will be smoother andsmoother. While conceptually the proof was straightforward, some ofthe technical details were formidable. In particular, we strongly usedthe compact support hypothesis on g . WELVE TALES 139 Our paper stimulated several others that obtained strengthening ofour result - two by Klein and others [84, 390] and one by March-Snitman[458]. Their techniques were very different from ours and each other.In particular, [390] only needs the weak condition on g that its Fouriertransform is C ∞ with all derivatives vanishing at ∞ (automatic if g is of compact support and the analog is even true if dκ is the Cantormeasure!).Besides these two major works on random potentials, I have paperson four other aspects ((1) and (3) only in one dimension). Let mebriefly discuss them.(1) localization for slowly decaying random potentials I wrote a num-ber of papers on the model (half or whole line) where a n ≡ b n = (1 + | n | ) − α ω n (11.19) where ω n are iirdv (sometimes with restrictions on their common distri-bution) [628, 147, 148, 383] and [665, Section 12.7]. The first and mostbasic paper [628] showed that if 0 < α < / 2, with minor assumptionson the distribution dκ , of ω (basically, it has the form (11.15) with g bounded and of compact support), then for a.e. ω , h ω has densepoint spectrum in [ − , 2] with eigenfunctions decaying at least as fastas e − C | n | β ; β = 1 − α . As noted there, the proof is an easy adaptationof the proof of localization in the one dimensional Anderson model byKunz-Souillard [402]. I pointed out that one knew (by the trace classtheory) there was pure a.c. spectrum on [ − , 2] when α > / ≤ α ≤ α < / α = 1 / g and added a coupling con-stant, λ in b n = λ (1 + | n | ) − / ω n . We showed for all sufficiently large λ , for a.e. ω , the model has dense point spectrum with power decayingeigenfunctions and for all sufficiently small λ no point spectrum. Sub-sequently Delyon [142] proved purely singular continuous spectrum inthese small λ regions.Kotani-Ushiroya [399] studied a closely related set of models. Theystudied 1D random continuum Schr¨odinger operators of the type stud-ied by the Russian group [244] but with the potential multiplied by(1 + | x | ) − α . They proved purely a.c. spectrum on [0 , ∞ ) when 1 / < α and sharpened the results of the last paragraph when α = 1 / 2. Kiselev,Last and I [383] used discrete analogs of Pr¨ufer variables to recover andstrengthen the results for the decaying discrete models. In particular, 40 B. SIMON if b n = λn − / ω n with ω n as in the classical Anderson model, we provedthat the spectrum is purely dense pure point if λ ≥ 12 and if λ < | E | ≤ (cid:112) − λ / − , α < 0. One might thinkthat | b n | → ∞ so the spectrum is discrete but if g has zero in itssupport, it might happen that although lim sup | b n | = ∞ , one has thatlim inf | b n | = 0! Indeed, when ω is uniformly distributed in [0 , − α > 1. When 0 < − α ≤ 1, thereis a semi-infinite interval of dense point spectrum.(2) Lifshitz tails I made some contributions to the theory of Lifshitztails [641, 377, 643] (I am embarrassed to say that I sometimes used theatypical spelling Lifschitz although I do note the original is Cyrillic).There is a huge literature, so I’ll only include my papers and the originalone of Lifshitz referring the reader to the excellent review of Kirsch-Metzger [376] from my 60th birthday festschrift for more referencesand more history. Here’s a rough heuristic argument close, to Lifshitzoriginal [448]. Consider a model H ω u ( n ) = 2 νu ( n ) − (cid:88) | j | =1 u ( n + j ) + b ω ( n ) u ( n ) ≡ [( H + b ω ) u ]( n ) (11.20) ? ? on Z ν with, say the b ω ( n ) uniformly distributed in [0 , H is a positive operator whosespectrum is [0 , ν ] and it is easy to prove that for a.e. ω the spectrum of H ω is [0 , ν + 1]. Imagine putting the system in a large box and lookingfor eigenvalues with energy less than ε and normalized eigenfunction ϕ .For (cid:104) ϕ, H ϕ (cid:105) to be small, ϕ must be spread out over a region of radius R at least ε − / with O( ε − ν/ ) sites. For (cid:104) ϕ, V ϕ (cid:105) to also be small, weneed V to be small at (most of) these sites, certainly no less than 1 / c ε − ν/ with 0 < c < 1. (Onecould argue that c should be ε but that only introduces a log term inthe exponent and would restrict the form of the single site probability). WELVE TALES 141 In any event, the expectation is that at leastlim E ↓ log( − log( k ( E ))) / log( E ) = − ν/ the weakest form of Lifshitz tails (and the only one that I proved). Theearly rigorous results in this area used the method of large deviations.My work was motivated by a breakthrough of Kirsch-Martinelli whofound the first proof that used bare hands rather than some fancyprobabilistic methods. They only obtained results of the form (11.21)(which was weaker than some earlier work) but for more general models.They relied on Dirichlet-Neumann bracketing [680, Section 7.5] andtreated continuum models. I wrote [641] mainly to advertise theirwork but also to extend it to the discrete case. The most importantcontribution of that paper was the use of Temple’s inequality whichwas often used in later works. In [377], Kirsch and I proved results like(11.21) for random perturbations of periodic problems near the bottomand top of the spectrum. We could not handle the issue of showingthere are also Lifshitz tails near the internal gap edges, a problem that,so far as I know, remains open, but I did handle the case of interior gapsin an Anderson model where there are gaps due to gaps in the supportof dκ [643]. I shouldn’t leave this subject without mentioning thereare interesting issues involving Lifshitz tails in random alloys with longrange potentials and in magnetic fields which are discussed in [376].(3) the notion of semi-uniform localization of eigenfunctions (SULE) Del Rio, Jitomirskaya, Last and I [138, 139] illuminated what expo-nential localization in random systems means (this work also discussedHausdorff dimension of singular continuous spectrum, so I will returnto it in Section 12). To use the title of our paper aimed to physicistswe dealt with the question, “What is localization?”. At the time wewrote it, given the acceptance of Anderson’s picture, many theoreticalphysicists would tell you that a system on Z ν is localized (at all ener-gies) means there is a complete set of eigenfunctions, { ϕ ω,m } ∞ m =1 , eachobeying | ϕ ω,m ( n ) | ≤ C ω,m e − A | n − n ω,m | (11.22) where n ω,m is the center of localization of the m th eigenfunction. Physi-cally though, localization means that a function which at time zero liveson a finite set should remain not too spread out uniformly in time. Thenatural estimate is to expect that E (cid:18) sup t | e − itH ω ( n, (cid:96) ) | (cid:19) ≤ Ce − ˜ A | n − (cid:96) | (11.23) 42 B. SIMON Indeed, Delyon et al [143] proved this for 1D Anderson models andAizenman [8] proved this in high dimension for large coupling Andersonmodels. One point of [138, 139] is that there are (non-random) modelswhere (11.22) holds but not only does (11.23) fails but in fact for any δ > (cid:104) e − itH δ , n e − itH δ (cid:105) /t − δ = ∞ ! In fact, it is just arank one perturbation (by cδ ) of the 1D AMO at coupling larger than2 with Diophantine coupling. Our point was that knowing how large C ω,m is critical for dynamic consequences of dense point spectrum. Onemight guess that one can take C independent of m but that doesn’thold in large classes of models. Instead, for a fixed H , we defined SULEto mean that for all δ > 0, there is C δ | ϕ m ( n ) | ≤ C δ e δ | n m | e − A | n − n m | (11.24) ? ? We proved that for operators H with simple spectrum this is equivalentto (and it general it implies)sup t | e − itH ( n, (cid:96) ) | ≤ C δ e δ(cid:96) e − A | n − (cid:96) | (11.25) We explicated the a.e. ω versions of this and noted that (11.23) implies(11.25) for a.e. ω .(4) simplicity of the spectrum in the localization regime In [647], Iproved that for a generalized Anderson model in arbitrary dimension,if, for a.e. ω , the spectrum is only dense pure point on an interval[ a, b ], then for a.e. ω and every n , δ n is cyclic for H ω (cid:22) [ a, b ], i.e. finitelinear combinations of { P [ a,b ] ( H ω ) H kω δ n } ∞ k =0 are dense in ran P [ a,b ] ( H ω ).In particular, this implies that the spectrum is simple on [ a, b ].Motivated in part by this, Jakˇsi´c-Last [328, 329], analyzed thesequestions more deeply. In particular they proved the result if “densepure point on an interval [ a, b ]” is replaced by “has spectrum on all of[ a, b ] with no a.c. part”. They needed a result of Poltaratskii on Hilberttransforms [512] to control singular continuous spectra and in [330],they provided a new proof of his result. Poltaratskii is a great expert onHilbert transforms, so when, in our study of consequences of Remling’swork, Zinchenko and I needed some facts about that transform, wejoined forces with Poltaratskii to prove what we needed [513].12. Generic Singular Continuous Spectrum (cid:104) s12 (cid:105) I like to joke that I spent the first part of my career proving thatsingular continuous spectrum never occurs (following Wightman’s “nogoo hypothesis” dictum) and the second part showing that it is generic!In 1978, Pearson [503] shocked most experts by showing that 1D con-tinuum Schr¨odinger operators with slowly decaying sparse potentials WELVE TALES 143 have purely singular continuous spectrum and, as discussed in Sec-tion 9, Avron and I proved that for suitable coupling and frequency,the AMO also had purely singular continuous spectrum but the phe-nomenon was still regarded as exotic and highly atypical. Starting inearly 1993, I discovered that, at least in the sense of Baire, it was, infact a generic phenomenon, a discovery sometimes called “the singularcontinuous revolution”. In the next few years I published eight pa-pers [648, 141, 343, 139, 687, 651, 652, 298] and two announcements[140, 138] on the subject. Later I studied the analog for OPUC [665,Section 12.4].I recall that the Baire category theorem [676, Theorem 5.4.1] saysthat a countable intersection of dense open sets in a complete metricspace is dense. Thus countable unions of nowhere dense sets (calledfirst category) are candidates for non-generic sets in that they are closedunder countable unions and their complements (the supersets of thedense G δ ’s) are dense. So dense G δ sets in complete metric spaces arecall Baire generic . Subsets of [0 , 1] of Lebesgue measure 1 are called Lebesgue generic . The notions can be radically distinct in that one canfind subsets A and B of [0 , 1] which are disjoint with one Baire genericand the other Lebesgue generic (and we will see shortly lots of spectraltheoretic cases where they are). We have already seen after (9.23) thatthe Diophantine rationals and the Liouville number provide such sets.One application of the Baire category theorem [676, Section 5.4] is forexistence. If a countable set of conditions each hold on a dense G δ ,they all hold somewhere (indeed on a dense G δ ). The most famousexample is an indirect proof of the existence of continuous, nowheredifferentiable functions [676, Problem 5.4.3].I should emphasize that the idea of s.c. spectrum being Baire genericunder some conditions was discovered before me by Sasha Gordon. Ina paper submitted in 1991 [246], he announced and in a paper [247]published about the same time as [141], Gordon found the same resultas in [141] (Theorem 12.2 below) with a different proof. Our work wasdefinitely later. I also note that in 1981, Zamfirescu [740] proved thesuggestive result that among all measures on [0 , 1] with a fixed boundedvariation and no pure points, which is a complete metric space in thevariation norm, a Baire generic measure is singular. See [676, Problem5.4.8] for a proof of the related result that if the probability measureson [0 , 1] is given the vague topology (in which it is a complete metricspace) a Baire generic one is purely singular continuous. I also note thatChoksi-Nadkarni in two papers [98, 99] (the first in 1990 predating mywork but which the authors say appeared in a (somewhat inaccessible)conference proceedings ) proved results for unitary operators analogous 44 B. SIMON to the results entitled Generic Self-Adjoint Operators below. Thatsaid, my presentation of the full panoply of situations with generic s.c.spectrum established the notion widely.My original motivation for this work involved a visit to Caltech byRaphael del Rio who gave a seminar on a result [136] related to thefollowing theorem which appeared in this form in the paper of del Rio,Makarov and Simon [141]: (cid:104) T12.1 (cid:105) Theorem 12.1 ([141]) . Consider a one parameter family of the form (11.1) where A is a bounded self-adjoint operator. Then there is a set, B ⊂ spec( A ) which is a dense G δ in spec( A ) so that no E ∈ B is aneigenvalue of any A λ ; λ (cid:54) = 0 . del Rio’s result was for boundary condition variation of Sturm-Liouville operators, only discussed a set being dense and uncountableand his proof was fairly involved but it had the key idea of studying theset we will define as B below and applying the Aronszajn-Donoghuetheory. Namely, we let B = { E | K ( E ) = ∞} (12.1) and proved that it was dense and uncountable. I was struck by thisresult which seemed surprising since, a priori, it certainly seemed pos-sible that for the Anderson model, the eigenvalues filled the entirespectrum as λ varied. Within a couple of days, I realized that dense G δ ’s were lurking and that there was a very short proof. For suppose E ∈ spec( A ) and there is an open interval, C , about E which is dis-joint from B so that G ( E ) < ∞ for all E ∈ C . It is easy to see thatthis condition implies that F ( E ) ≡ lim ε ↓ F ( E + iε ) exists and is real.Since the imaginary part vanishes E / ∈ L ∪ S which, by Theorem 11.2,is dense in spec( A ). It follows that E / ∈ spec( A ) contradicting that E ∈ spec( A ). We conclude that B is dense in spec( A ). Moreover, by asimple argument, G is lower semicontinuous, so { E | G ( E ) > n } is openand thus B is a G δ . By Theorem 11.2(c), no E ∈ B is an eigenvalue ofsome A λ ; λ (cid:54) = 0.Much more interesting than the set of forbidden energies is the setof forbidden coupling constants and, in this regard, one has (cid:104) T12.2 (cid:105) Theorem 12.2 (Gordon[247], delRio-Makarov-Simon[141]) . Considera one parameter family of the form (11.1) where A is a bounded self-adjoint operator. Then { λ | A λ has no eigenvalues in spec( A ) } is adense G δ in R . The proof relies on the fact that on the set P where K ( E ) < ∞ , theboundary value, F ( E ), exists and is real, and by Theorem 11.2(c), the WELVE TALES 145 corresponding λ ’s are given by F ( E ) = − λ − . Since λ (cid:55)→ − λ − takescountable unions of closed nowhere dense subsets of ± (0 , ∞ ) to suchunions, it suffices to write P as a union of such sets each of which ismapped by F to such a set. One does this by finding such a union sothat on each such set, F is Lipschitz. One needs to go into the complexplane to do this.This result implies the remarkable fact that in the Anderson model,if you fix a random choice of potential at all site sites but one, thenfor a Lebesgue generic choice at the last point, the spectrum is entirelypure point while for a different Baire generic choice it is purely singularcontinuous. In particular, dense pure point spectrum will turn into sin-gular continuous spectrum under some arbitrarily small perturbations.But there is an asymmetry. In the context of general rank one pertur-bations, if there is dense pure point spectrum, there is always a denseset of couplings with purely singular continuous spectrum, but thereare examples where for all coupling, the spectrum is purely singularcontinuous.This next step on forbidden coupling constant was quite natural andshortly after I figured out Theorem 12.1, del Rio and I started workingof what became Theorem 12.2 inviting Makarov to join us when weran into difficulty. As we were doing that, I asked myself if there mightnot be a general mechanism underlying this phenomenon of genericsingular continuous spectrum and I realized that the key was some softanalysis. I found the following (cid:104) T12.3 (cid:105) Theorem 12.3 ([648]) . Let A be a family of self-adjoint operators on aHilbert space, H , which is given a metric topology in which convergenceimplies strong operator convergence of resolvents and in which A is acomplete metric space. Then the following sets are all G δ sets: (a) For each closed set, C ⊂ R , the set of A ∈ A with no eigenvaluesin C . (b) For each open set, U ⊂ R , and each fixed vector ψ ∈ H , the set of A ∈ A so that the spectral measure obeys (cid:16) µ ( ψ ) A (cid:17) ac [ U ] = 0 . (c) For each closed set, K ⊂ R , the set of A ∈ A with K ⊂ spec( A ) . Remarks. 1. While my proof is not hard, it is a little awkward andunnatural. Lenz-Stollmann [428] found a more natural and direct proofand also slight extensions where rather than putting a topology on A ,one looked at continuous images of complete metric spaces and, in[649], I provided a different simplification of the proof.2. I emphasize that this theorem says nothing about density. Thatmay or may not hold. 46 B. SIMON I singled out one consequence of this because it has such an Alice-in-Wonderland character: ? (cid:104) T12.4 (cid:105) ? Theorem 12.4 (The Wonderland Theorem [648]) . Let A be a familyof self-adjoint operators on a Hilbert space, H , which is given a metrictopology in which convergence implies strong operator convergence ofresolvents and in which A is a complete space. Suppose that a denseset in A has purely a.c. spectrum on an open interval I ⊂ R andanother dense set has purely dense point spectrum on I . Then fora dense G δ of A ∈ A , I lies in spec( A ) and the spectrum is purelysingular continuous there! Remarks. 1. I stated the result this way for drama but the sameconclusion holds under the weaker hypothesis that there is a densesubset with no eigenvalues in I and another with no a.c. spectrum in I and another with I ⊂ spec( A ).2. The proof is immediate from Theorem 12.3. If I = ( a, b ) let C n be the set of A ∈ A in Theorem 12.3(a) when K = [ a + 1 /n, b − /n ].If { ψ m } ∞ m =1 is an dense set in H , let K m be the set of A ∈ A inTheorem 12.3(b) when U = I and ψ = ψ m . Finally let P k be the set of A ∈ A with [ a + 1 /k, b − /k ] ⊂ spec( A ). By the hypothesis, all thesesets are dense and by Theorem 12.3 all are G δ sets. So, by the Bairecategory theorem, their intersection is a Baire generic set. If A is intheir intersection, it has I ⊂ spec( A ) and there are no eigenvalues in I and no a.c. spectrum.Here are some of the applications of this set of ideas:(1) Generic Self-Adjoint Operators Let A be the set of all self-adjoint operators, A with (cid:107) A (cid:107) ≤ A ∈ A has spectrum all of [ − , 1] and purely singular continu-ous spectrum! As mentioned above, Choksi-Nadkarni [98] hadproven the same result for unitary operators and later, theynoted that using Cayley transforms, their results imply the re-sult for self-adjoint operators. They are also noted that gener-ically, the spectrum is simple (which is easy to prove with asimple “this set is a G δ argument” and the fact that small per-turbations of operators with dense point spectrum are operatorswith simple dense point spectrum).The proofs depend on a result of Weyl [731] (see [680, Theo-rem 5.9.2]) that any self-adjoint operator is a norm limit of op-erators with point spectrum and then a short argument that one WELVE TALES 147 can norm approximate operators with dense point spectrum byones with purely a.c. spectrum. This result illuminates Weyl’s.(2) A Generic Weyl-von Neumann Theorem Let I p be the traceideal with (cid:107)·(cid:107) p norm ([680, Section 3.7] or [662]). von Neumann[722] extended Weyl’s result to allow small Hilbert-Schmidt ( I )perturbations and Kuroda [404] allowed arbitrarily small I p forany p > p = 1 by the Kato-Rosenbum Theorem[527, Theorem XI.8]). Using this, I proved that for any self-adjoint operator A , and any p > 1, for a (cid:107)·(cid:107) p -topology Bairegeneric B ∈ I p , the spectrum of A + B on spec ess ( A ) is purelysingular continuous.(3) Generic Discrete Schr¨odinger Operators with Bounded Poten-tial Let Ω = × ∞ n = −∞ [ α, β ] with the product topology and given ω ∈ Ω let A ( ω ) by the Jacobi matrix with all a n = 1 and b n = ω n . For a dense G δ , A ( ω ) has purely s.c. spectrum. It iseasy to see this: because of Anderson localization, the set of ω whose spectrum is [ − α, β ] and pure point is dense in Ω(and a G δ by Theorem 12.3). Moreover, given any ω , it is easyto see it a limit of weak of periodic ω ’s so the set ω for which A ( ω ) has only a.c. spectrum is dense. One concludes that aBaire generic A ( ω ) has spectrum [ − α, β ] and purelysingular continuous spectrum.(4) Generic Schr¨odinger Operators with Slowly Decaying Potential In [648], I proved if you look at C ∞ ( R ν ), the continuous func-tions on R ν vanishing at ∞ , then, Baire generically, − ∆ + V has purely singular continuous spectrum on [0 , ∞ ). The con-tinuous functions of compact support for which the spectrumof the associated Schr¨odinger operator is pure a.c. on [0 , ∞ ) isdense. Moreover, by the results of Deift-Simon [133], the a.c.spectrum is unchanged by modifying V inside a finite ball, so ifone finds a V ∈ C ∞ ( R ν ) with associated Schr¨odinger operatorhaving no a.c. spectrum, given an W , we find V n ∈ C ∞ ( R ν )equal to W on the ball of radius n and V outside the ball ofradius n + 1, to show that the V ’s whose associated Schr¨odingeroperator has no a.c. spectrum is dense. One finds the required V by taking a centrally symmetric decaying random potentialand using [628].(5) Generic Discrete D Schr¨odinger Operators with Slowly Decay-ing Potential Using similar ideas, [648] fixes α ∈ (0 , / 2) andlooks at Jacobi matrices with a n ≡ b n inthe space of sequences so that | n + 1 | α b n → 48 B. SIMON metric space with norm sup n ( | n + 1 | α | b n | ) to get a Baire genericfamily of Jacobi matrices with purely singular continuous spec-trum in ( − , α < / ? (cid:104) T12.5 (cid:105) ? Theorem 12.5 (Jitomirskaya-Simon [343]) . Let H ω be a discreteSchr¨odinger operator of the form (9.1) . Suppose that for some ω ∈ Ω b n ( ω ) is even in n (for example the AMO, (9.3) ). Then for a dense G δ , U ⊂ Ω , H ω has no eigenvalues if ω ∈ U . Remarks. 1. In particular, if one knows that there is no a.c. spectrum(when [343] was written, the result of Kotani [396] and Last-Simon [415]that the a.c. spectrum of H ω was constant was not known, so instead, we noted that so long as it was known that there was at least one ω with no a.c. spectrum, one had it for a dense G δ ), then for a Bairegeneric ω , H ω has purely s.c. spectrum. In particular, this is true forthe AMO when λ > 2. So when the frequency is Diophantine, we havea situation where for a Lebesgue generic θ , the spectrum is dense purepoint and for a Baire generic theta , the spectrum is purely singularcontinuous!2. We needed the condition that some function in the hull is even touse Gordon’s lemma which is discussed after (9.23).3. A year later, motivated by this results, Hof, Knill and I [298]proved generic s.c. spectrum for a class of subshift potentials whichwhile not strictly almost periodic or even are closely related. See [298]for details or Damanik [119] for more on subshifts.Besides these four papers, I had several additional papers in the series Operators with singular continuous spectrum . The most substantial,indeed, by far the longest paper in the series is with del Rio, Jito-mirskaya and Last [139] (announced in [138]). As already mentionedin Section 11, that paper discussed localization for random quantumsystems (see point (3) in the discussion including (11.22)) but its mainfocus involves the Hausdorff dimensions of the support of the singularcontinuous spectral measures for Anderson Hamiltonians and of the WELVE TALES 149 set of λ in Theorem 12.2 (for more on Hausdorff dimension, see Fal-coner [180] and for Hausdorff dimension of measure, see Rogers [539]or Simon [676, Section 8.2]).In the context of general rank one perturbation theory, the set of λ leading to singular continuous spectrum can be large, e.g. if the initialmeasure is pure point but the set P of (11.6) is empty (which canhappen for suitable initial measure dµ ), then, by Theorem 11.2, thereis purely singular continuous spectrum for all λ (cid:54) = 0. A major pointof [139] is that when one has SULE (see (11.22)), the complement of P , i.e. the set, B , of E where G ( E ) = ∞ has Hausdorff dimension 0.Using this one finds: ? (cid:104) (cid:105) ? Theorem 12.6 (delRio et al. [139]) . Consider a generalized Ander-son model with SULE. Then for a.e. choice of potential, if we varythe potential at a single point, say b , then the set of such b whichhave any singular continuous spectrum has Hausdorff dimension zero.Moreover, for any such value, the spectral measure is supported ona set of Hausdorff dimension zero and one has that for t large, that (cid:104) δ , x ( t ) δ (cid:105) ≤ C (log | t | ) . The last three papers in the series are addenda to the main themes.Paper 5 with Stolz [687] has nothing to do with Baire genericity. Ratherit has criteria for sparse potentials to have no point spectrum, so, ifone can also assure no a.c. spectrum, the spectrum is pure s.c. Forthe examples of Pearson [503], this provides an independent proof ofthe absence of point spectrum. By getting no a.c. spectrum with themethod of Simon-Spencer [686] (see Theorem 6.21), one gets explicitexamples with purely singular continuous spectrum. Paper 6 [651]presented the first examples of graph Laplacians and Laplace Beltramioperators on manifolds with purely singular continuous spectra. Paper7 [652] constructed a multidimensional example with high barriers asin Simon-Spencer [686] but still having a.c. spectrum. This mightseem to have nothing to do with s.c. spectrum but the example isseparable, built of two 1D operators which have s.c. spectrum, but withtime decay one can compute and then use to prove the sum has a.c.spectrum (my work was in part motivated by an unpublished remarkof Malozemov and Molchanov that because the convolution of two s.c.measures can be a.c., it might be possible to construct examples likethis). 50 B. SIMON Further Remarks ? (cid:104) s13 (cid:105) ? As I indicated earlier, while there have been references to some workafter that cutoff, this paper mainly discusses research done before 1995.My research since had its roots in the earlier work but went in a di-rection which seems to have less relevance to quantum physics. Asdiscussed in Section 6, H = − ∆ + V ( x ) has “normal” spectral behav-ior if V ( x ) decays faster than | x | − , namely only discrete spectrum in( −∞ , 0) and purely a.c. spectrum on (0 , ∞ ). On the hand as I dis-cuss in Sections 11 and 12, if the decay is slower than | x | − / , one cansometimes have no a.c. spectrum. This was realized by 1995.I began to wonder what happen for decay like | x | − α with 1 / < α < 1. If V has a gradient decaying faster than | x | − , it was known formany years (see, e.g. [729]) that one has “normal” spectral behaviorand in the random case, one also has this (see, e.g. [399]). Whatcould happen in the general case? Fortunately, at this time, I had twovery talented grad students - Sasha Kiselev from St. Petersburg andRowan Killp from Auckland in New Zealand. Interestingly enough,they both came to work with me upon the strong recommendation ofBoris Pavlov [403] who moved from St. Petersburg to Auckland at atime to interact with each of them as undergraduates. I gave Kiselevthe problem of whether there was always a.c. spectrum for | x | − α decaywhen 1 / < α < / < α < / < α < V or b in L ( R , dx ), a problem which caught Killip’s fancy. Percy Deift visitedCaltech and when Killip told him about the conjecture, given Percy’swork on exactly integrable systems, he immediately thought about thesum rule of Gardner et al [211] that for any nice enough potential, V ( x ), on R , one has that (cid:90) ∞−∞ V ( x ) dx = 163 (cid:88) j | E j | / + 8 π (cid:90) ∞−∞ log (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) T ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) k dk (13.1) where, say, V is bounded with compact support, { E j } are the negativeeigenvalues of − d dx + V ( x ) and T ( k ) is the transmission coefficient atenergy E = k . Interestingly enough, more than 20 years before, in[447], Lieb and Thirring had dropped the last term in (13.1) to get aLieb-Thirring inequality which they could prove was optimal, becausefor soliton potentials, the dropped term vanishes. Deift and Killip [132] WELVE TALES 151 dropped the middle term of (13.1) and got an inequality they could useto prove that if V ∈ L (0 , ∞ ) (resp b ∈ (cid:96) ( Z + )), then H = − d dx + V (resp. hu ( n ) = u ( n + 1) + u ( n − 1) + b n u ( n )) has a.c. spectrum [0 , ∞ )(resp [ − , Do there exist potentials V ( x ) on [0 , ∞ ) so that | V ( x ) | ≤| x | − / − ε for some ε > and so that − d /dx + V ( x ) has some singularcontinuous spectrum . At the time, I didn’t realize that an analogousproblem had been solved in 1936 by Verblunsky [721]! For the analog ofthe potential for OPUC are what are now called Verblunsky coefficients(a term I introduced in 2005), a sequence { α n } ∞ n =0 of numbers in D associated to any probability measure, dν on ∂ D . What Verblunskyproved, extending a result of Szeg˝o [706, 674], is that this sequence liesin (cid:96) if and only if the a.c. part of dν obeys a certain condition. Thesingular part could be arbitrary so long as the total mass of dν was 1.These ideas were brought into the question of mixed spectrum forSchr¨odinger operators by Sergui Denisov, then a graduate student inMoscow, in a preprint I first learned about in January 2001 (a ver-sion only appeared in print several years later [152]). Nick Makarovand I were impressed enough that we invited him to be a postdocat Caltech, of which more shortly. Denisov used a continuum ana-log of OPUC called Krein systems and said that he could construct V ∈ L ((0 , ∞ ) , dx ) so the corresponding H had an arbitrary singularcontinuous part on some [0 , E ]. Technically that didn’t solve my prob-lem since I had stated the result in terms of power decay, not L p (thepower result result was obtained shortly afterwards by Kiselev [382])but to me morally it did.Rowan Killip and I set out to understand Denisov’s proof but, inpart, because we had no prior experience with OPUC or Krein systemsand could find little literature on the former and none on the later,we found the arguments opaque. We did determine that the key tohis proof seemed to be sum rule. We were interested in a result forJacobi matrices, so we looked at some sum rules of Case [95]. His sumrules were only formal so it wasn’t clear when they hold although hisarguments certainly could be made rigorous if b n and a n − 52 B. SIMON We found that none of Case sum rules (which entered from succes-sive terms in a Taylor series) had the necessary positivity but by foolingaround, we found a linear combination of two of them that was positive.It was mysterious why there was any such combination and the rathercomplicated functions that entered in the final sum rule were totallyadhoc. Fifteen years later Gamboa, Nagel and Roault [209] found a to-tally new proof using the method of large deviations on certain randommatrix ensembles that explained why there was a positive quantity andthe meaning of the previously ad hoc functions. For the OPUC analog,Breuer, Zeitouni and I found some other positive sum rules using largedeviations [80]. Since the GNR paper wasn’t very accessible to spectraltheorists, we wrote a pedagogic exposition of their approach [79].In going from the finite support case to the general, there was onetricky limit that stymied us for a while. In those days, jury duty in LosAngeles could mean coming in every day for two weeks waiting in thejury assembly room all day for assignment to a trial and in the sum-mer of 2001, I had such a stint not even winding up on a jury! Sittingaround gave me lots of time to think about this holdup and I realizedthat since the object that we were having having trouble with was a rel-ative entropy, it had some semicontinuity properties that overcame ourdifficult. We were quite pleased by this discovery although we learnedseveral years later that Verblunsky [721] had made the same discovery65 years earlier in his related work! (Verblunsky didn’t know he hadan entropy but he had discovered and exploited the semicontinuity.)One result of my work with Killip [374] was necessary and sufficientconditions on a spectral measure for associated Jacobi matrix to have J − J a Hilbert Schmidt operator (where J corresponds to a n = 1 , b n =0), a result now regarded as an OPRL analog of Szeg˝o’s theorem. Wealso obtained results described in (8.20) which led to a proof of Nevai’sconjecture.While we writing this up, Denisov arrived at Caltech and gave acourse on Krein systems which I sat in on. What struck me was hisinitial few works where he described the theory of OPUC. I was struckby the beauty and elegance of the subject although I found a muchsimpler proof of Szeg˝o recursion than he gave us (it turned out theproof that I’d found was in the literature but so little known that Isurprised at least one expert with it). It became clear to me that thesimilarities between OPUC and OPRL suggested one should be able tocarry over (often with some gymnastics needed) much of the spectraltheory of Jacobi matrices to OPUC. Rather than lots of small paperson each subarea, I decided one long review article made sense. I hadto face the fact that there wasn’t really any recent exposition of the WELVE TALES 153 basics of OPUC so I decided on a two long review articles which growinto a two volume set of books [664, 665] with over 1000 pages!In many ways, the spectral theory of OPUC and its relation toOPRL became a major focus of my research for the time since. Manyof the major results involve Szeg˝o’s theorem and Szeg˝o asymptotics,among them the extension by Damanik-Killip-Simon [122] of the workof Killip-Simon to perturbations of certain periodic Jacobi matrices,my study with Christiansen and Zinchenko on Szeg˝o behavior of fi-nite gap operators [101, 102, 103] and the work with Damanik [123] onnecessary and sufficient conditions for Szeg˝o asymptotics for OPRL.OPRL can be viewed as the solutions of an L minimization problem.The analogous L ∞ minimization problem define Chebyshev polynomi-als which depend on some compact subset, e ⊂ C . In a brilliant paper1969 paper, Widom [733] discussed how to modify Szeg˝o asymptoticsfor Chebyshev polynomials when e is a finite union of disjoint suf-ficiently smooth Jordan curves. He obtained partial results for finitegap sets in R and he made a conjecture about the expected asymptoticswhich we dubbed Szeg˝o-Widom asymptotics. This conjecture remainedopen for over 45 years until proven by Christensen, Zinchenko and me[104]. Our proof could be phrased in terms of discriminants of periodicJacobi matrices which made the arguments natural.It is appropriate to end with a story about the publication of thatpaper. I felt the paper was important enough to warrant sending it to atop three journal but, for various reasons, one of my coauthors wantedto send it to a slight less prestigious but still top journal. Fairly quicklywe got a reply that the person asked for a quick opinion thought it wasnice that we had solved a 45+ year old conjecture but the paper wasn’tup to the standard of this journal because the proofs were too simple!I was scandalized by this and insisted that we try a top three journal,which we did, where the paper was accepted. Lest you think I disap-prove of the system that top math journals use to decide which papersto publish, I feel that it is best described by Churchill’s description ofdemocracy - the worst possible method of evaluation except for all theothers. References AdamsGpRep [1] J.F. Adams, Lectures on Lie Groups , W.A. Benjamin, New York, 1969. DamOpen [2] F. Adiceam, D. Damanik, F. G¨ahler, U. Grimm, A. Haynes, A. Julien,A. Navas, L. Sadun and B. Weiss, Open problems and conjectures related tothe theory of mathematical quasicrystals , Arnold Math. J. (2016), 579–592. AgmonAC [3] S. 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Simon, Pointwise bounds on eigenfunctions and wave packets in N-bodyquantum systems, III , Trans. AMS (1975), 317–329. SimonCI2 [599] B. Simon, Correlation inequalities and the mass gap in P ( ϕ ) , II:Uniquenessof the vacuum for a class of strongly coupled theories , Ann. of Math. (1975), 260–267. SimonVanc [600] B. Simon, Approximation of Feynman integrals and Markov fields by spin sys-tems , in Proceedings of the International Congress of Mathematicians . Volume2. Held in Vancouver, B. C., August 21–29, 1974. Canadian MathematicalCongress, Montreal, Que., 1975, pp 399–402. SiDiam [601] B. Simon Universal diamagnetism of spinless bose systems , Phys. Rev. Lett. (1976), 1083–1084. SiWkCp [602] B. Simon, The bound state of weakly coupled Schr¨odinger operators in oneand two dimensions , Ann. Phys. 97 (1976), 279-288 SiGenericGaps [603] B. Simon, On the genericity of nonvanishing instability intervals in Hill’sequation , Ann. Inst. H. Poincar´e A24 (1976), 91–93. WELVE TALES 183 SiWkTrace [604] B. Simon, Analysis with weak trace ideals and the number of bound states ofSchr¨odinger operators , Trans. AMS. (1976), 367–380. SiBargBdSt [605] B. Simon, On the number of bound states of two-body Schr¨odinger operators:A review , in [445, pp 305-326]. SiBargQDyn [606] B. Simon, Quantum dynamics: From automorphism to Hamiltonian , in [445,pp 327-349]. SimonScatteringQF [607] B. Simon, Scattering theory and quadratic forms: on a theorem of Schechter ,Comm. Math. Phys. (1977), 151–153. SimonHVZ [608] B. Simon, Geometric methods in multiparticle quantum systems , Comm.Math. Phys. (1977), 259–274. SimonKI1 [609] B. Simon, An abstract Kato’s inequality for generators of positivity preservingsemigroups , Indiana Univ. Math. J. (1977), 1067–1073. SimonLSCForms [610] B. Simon, Lower semicontinuity of positive quadratic forms , Proc. Roy. Soc.Edin. (1977), 267–273. SiLinear [611] B. Simon, On the absorption of eigenvalues by continuous spectrum in regularperturbation problems , J. Func. Anal. (1977), 338–344. SimonMonotoneForms [612] B. Simon, A canonical decomposition for quadratic forms with applications tomonotone convergence theorems , J. Func. Anal. (1978), 377–385. SimonCSRev [613] B. Simon, Resonances and complex scaling: A rigorous overview , Intl. J.Quant. Chem. (1978), 529–542. SimonMaxMinForm [614] B. Simon, Maximal and minimal Schr¨odinger forms , J. Op. Th. (1979),37–47. SimonExtCS [615] B. Simon, The definition of molecular resonance curves by the method ofexterior complex scaling , Phys. Lett. (1979), 211–214. SimonEnss [616] B. Simon, Phase space analysis of simple scattering systems: Extensions ofsome work of Enss , Duke Math. J. (1979), 119–168. SiCLQSA [617] B. Simon, Identifying the classical limit of a quantum spin system , Colloq.Math. Soc. Bolyai (1979), 989–1001. SimonKI2 [618] B. Simon, Kato’s inequality and the comparison of semigroups , J. Func. Anal. (1979), 97–101. SiCLQS [619] B. Simon, The classical limit of quantum partition functions , Comm. Math.Phys. (1980), 247–276. SimonLSAnon [620] B. Simon, Decay of correlations in ferromagnets , Phys. Rev. Lett. (1980),547–549. SimonLS [621] B. Simon, Correlation inequalities and the decay of correlations in ferromag-nets , Comm. Math. Phys. (1980), 111–126. SimonUBMF [622] B. Simon, Mean field upper bound on the transition temperature in multicom-ponent ferromagnets , J. Stat. Phys. (1980), 491–493. SiCritical [623] B. Simon, Brownian motion, L p properties of Schr¨odinger operators and thelocalization of binding , J. Func. Anal. (1980), 215–229. SiHTDecay [624] B. Simon, The rate of falloff of Ising model correlations at large temperature ,J. Stat. Phys. (1981), 53–58. SimonCritical1D [625] B. Simon, Absence of continuous symmetry breaking in a one-dimensional n model , J. Stat. Phys. (1981), 307–311. SimonSchSmgp [626] B. Simon, Schr¨odinger semigroups , Bull. Amer. Math. Soc. (1982), 447–526. SiAPFlu [627] B. Simon, Almost periodic Schr¨odinger operators: A review , Adv. Appl. Math. (1982), 463–490. 84 B. SIMON SiRPt [628] B. Simon, Some Jacobi matrices with decaying potential and dense point spec-trum , Comm. Math. Phys. (1982), 253–258. SiCont [629] B. Simon, Continuity of the density of states in magnetic field , J. Phys. A15 (1982), 2981–2983. SiKotani [630] B. Simon Kotani theory for one dimensional stochastic Jacobi matrices , Com-mun. Math. Phys. 89 (1983), 227-234. SiLorentz [631] B. Simon, Equality of the density of states in a wide class of tight bindingLorentzian models , Phys. Rev. B27 (1983), 3859–3860. SiBerry [632] B. Simon Holonomy, the quantum adiabatic theorem and Berry’s phase , Phys.Rev. Lett. (1983), 2167–2170. SiSC1 [633] B. Simon, Semiclassical analysis of low lying eigenvalues, I. Non-degenerateminima: Asymptotic expansions , Ann. Inst. H. Poincar´e (1983), 295–307. SiSCAnon [634] B. Simon, Instantons, double wells and large deviations , Bull. AMS (1983),323–326. SiNQC1 [635] B. Simon, Some quantum operators with discrete spectrum but classically con-tinuous spectrum , Ann. Phys. (1983), 209–220. SiNQC2 [636] B. Simon, Nonclassical eigenvalue asymptotics , J. Func. Anal. (1983), 84–98. SiSC2 [637] B. Simon, Semiclassical analysis of low lying eigenvalues, II. Tunneling , Ann.Math. (1984), 89–118. SiSC3 [638] B. Simon, Semiclassical analysis of low lying eigenvalues, III. Width of theground state band in strongly coupled solids , Ann. Phys. (1984), 415–420. SiMaryland [639] B. Simon, Almost periodic Schr¨odinger operators, IV. The Maryland model ,Ann. Phys. (1985), 157–183. SiSC4 [640] B. Simon, Semiclassical analysis of low lying eigenvalues, IV. The flea on theelephant , J. Func. Anal. (1985), 123–136. LifSi1 [641] B. Simon, Lifshitz tails for the Anderson model , J. Stat. Phys. (1985),65–76. SiLoc1 [642] B. Simon, Localization in general one dimensional random systems, I. Jacobimatrices , Comm. Math. Phys. (1985), 327–336. LifSi2 [643] B. Simon, Internal Lifschitz tails , J. Stat. Phys. (1987), 911–918. SiBall2 [644] B. Simon, Absence of ballistic motion , Comm. Math. Phys. (1990) 209–212. SimonJellyRoll [645] B. Simon The Neumann Laplacian of a jelly roll , Proc. Amer. Math. Soc. (1992), 783–785. SimonLG [646] B. Simon, The Statistical Mechanics of Lattice Gases , Princeton UniversityPress, 1993. SimonCyc [647] B. Simon, Cyclic vectors in the Anderson model , Rev. Math. Phys. (1994),1183–1185. SiSingC1 [648] B. Simon, Operators with singular continuous spectrum: I. General operators ,Ann. of Math. (1995), 131–145. SimonLpDecomp [649] B. Simon, L p norms of the Borel transform and the decomposition of mea-sures , Proc. AMS (1995), 3749–3755. SiVanc [650] B. Simon, Spectral analysis of rank one perturbations and applications , in Mathematical quantum theory. II. Schr¨odinger operators (Vancouver, BC,1993) , ed. J. Feldman, R. Froese and L. Rosen, pp. 109–149, CRM Proc.Lecture Notes, , Amer. Math. Soc., Providence, RI, 1995. WELVE TALES 185 SiSingC6 [651] B. Simon, Operators with singular continuous spectrum, VI. Graph Laplaciansand Laplace-Beltrami operators , Proc. AMS (1996), 1177–1182. SiSingC7 [652] B. Simon, Operators with singular continuous spectrum, VII. Examples withborderline time decay , Comm. Math. Phys. (1996), 713–722. SimonGpRep [653] B. Simon, Representations of Finite and Compact Groups , Graduate Studiesin Mathematics , American Mathematical Society, 1996. SimonACBdd [654] B. Simon, Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schr¨odinger operators , Proc. Amer. Math. Soc. (1996),3361–3369. SimonDPS [655] B. Simon, Some Schr¨odinger operators with dense point spectrum , Proc.Amer. Math. Soc. (1997), 203—208. SimonSpecAv [656] B. Simon, Spectral averaging and the Krein spectral shift , Proc. 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critical ⇐⇒ − ∆ + (1 + ε ) V is supercritical for all ε > − ∆ + V ) = 0Thus critical V ’s are ones that like − ∆ in 1 and 2 dimensions areabout to give birth to bound states. A main result of [623] is that V is subscritical ⇐⇒ sup t (cid:107) e − ( − ∆+ V ) t (cid:107) ∞ , ∞ < ∞ (the norm from L ∞ to L ∞ ). Both [623] and [386] show that if V and W are both criticalin three dimensions, then − ∆ + V ( x ) + W ( x − R ) has a bound stateof energy, E ( R ) < R with E ( R ) ∼ − βR − . [386] evencomputes the universal value of β . This result is connected to theEffimov effect. A considerable literature has developed on the study ofcritical operators; [508] reviewed the literature as of 2005.In [387, 388], Klaus and I consider the variety of coupling constantthreshold behavior that can occur for − ∆ + V + λW (with V and W short range) when as λ ↓ λ , some eigenvalue is absorbed in thecontinuous spectrum (the first paper deals with the two body problemand the second with a limited set of N body systems). The results arequite complicated and supplement/illuminate some work of Jensen-Kato [334].Our next topic concerns the situation where V goes to infinity atinfinity (or, at least, is bounded away from zero there) but min( V ( x )) =0 and we are interested in the lowest eigenvalues of − ∆+ λ V as λ → ∞ .In [633] I proved (cid:104) T8.6 (cid:105)