aa r X i v : . [ m a t h . SP ] J u l SCHR ¨ODINGER OPERATORS WITH THIN SPECTRA
DAVID DAMANIK AND JAKE FILLMAN
Abstract.
The determination of the spectrum of a Schr¨odingeroperator is a fundamental problem in mathematical quantum me-chanics. We discuss a series of results showing that Schr¨odingeroperators can exhibit spectra that are remarkably thin in the senseof Lebesgue measure and fractal dimensions. We begin with a briefdiscussion of results in the periodic theory, and then move to a dis-cussion of aperiodic models with thin spectra. Introduction
The purpose of this text is to discuss some recent work onSchr¨odinger operators with thin spectra. We will begin with a listof motivating questions and then focus on one of them, perhaps themost fundamental among them, and progressively ask more and moredetailed refinements of this question, each of them prompted by a dis-cussion of the previous one. This progression will lead us to the cur-rent state of affairs, where Schr¨odinger operators with surprisingly thinspectra have been identified, likely somewhat at odds with the expec-tations one might have had based on intuition or classical paradigms.This note is not meant to be a comprehensive survey, and the ques-tions we ask and the results we present will be selected according tothe goal outlined in the previous paragraph: motivating and explainingthe recent results on Schr¨odinger operators with thin spectra. For addi-tional references and a discussion of the historical context we refer thereader to the original articles. Throughout the text, where appropriate,we point the reader towards suitable survey articles and textbooks foradditional reading.
Date : July 6, 2020.D.D. was supported in part by NSF grant DMS–1700131 and by an Alexandervon Humboldt Foundation research award.J.F. was supported in part by a Simons Collaboration Grant. Schr¨odinger Operators and Things we Might Want toKnow About Them
Schr¨odinger operators are central to the mathematical formulationof quantum mechanics, indeed their associated unitary group generatesthe time evolution of a quantum mechanical system. We will focus onthe simplest case of a single (quantum) particle exposed to a potential V and ignore all physical constants. Doing so we arrive at operators ofthe form H = H V ,(2.1) H = − ∆ + V in H = L ( R d ) , where ∆ is the standard Laplacian ,(2.2) [∆ ψ ]( x ) = d X j =1 ∂ ψ∂x j ( x ) , and V acts by multiplication with the potential V : R d → R (denotedby the same symbol, as is customary),(2.3) [ V ψ ]( x ) = V ( x ) ψ ( x ) . We will refer to d as the dimension . Its value is of crucial importance;there is a huge difference between the cases d = 1 and d >
1, bothin terms of the results one should expect to hold and the methodsthat exist to establish these results. Even in the higher-dimensionalcase d >
1, the value of d is sometimes of importance, and theremay be further transitions in the expected or provable behavior as d is increased.It is often beneficial to initially consider the free case where V van-ishes identically, so that one deals with − ∆, and to only later add onthe potential, essentially as a perturbation of − ∆.Naturally one cannot define (2.2) on all of H , at least if one wantsthat object to belong to H as well. Thus, it is an important problem toidentify a suitable domain of this operator. It should not be too large(so that − ∆ does not map some elements of the proposed domain out-side H ) and it should also not be too small (for reasons that becomeimportant later: we want to have a well defined time evolution e − itH ).This leads to the goal of choosing the domain such that the operatoris self-adjoint . Once this has been accomplished for the free case, thesame domain works for large classes of potentials V , and hence thetwo-step procedure works out neatly in these cases. We won’t dwell onthis particular issue further, referring the reader instead to [50], and CHR ¨ODINGER OPERATORS WITH THIN SPECTRA 3 will assume henceforth that self-adjointness will always have been ad-dressed successfully for the Schr¨odinger operators H we consider. In asimilar vein, most theorems may be formulated for various classes suchas continuous potentials, smooth potentials, uniformly locally square-integrable potentials, and could even be pushed beyond that to evenweaker regularity classes; to keep the exposition free of technicalities,we will mostly suppress assumptions on the regularity of V , and wewill not attempt to state theorems in their absolute most general form.The reader is welcome to suppose that all potentials are bounded andcontinuous unless they are explicitly said to be otherwise.Having realized (2.1) as a self-adjoint operator in H , we can inferthat the spectrum σ ( H ) must be real. We will comment on the impor-tance of this set momentarily. Let us first note that several interestingdecompositions of this set exist. The most basic decomposition is σ ( H ) = σ disc ( H ) ∪ σ ess ( H ) , where the discrete spectrum σ disc ( H ) consists of all eigenvalues of H that have finite multiplicity, and the essential spectrum is the comple-mentary set. In particular, this is a disjoint union, that is, a partitionof the spectrum. By definition, and the fact that L ( R d ) is separable,the discrete spectrum is a countable set, and it is therefore irrelevantfor the questions we will discuss here – whether the spectrum is “thin.”Another important aspect, which however is less directly related to ourtheme, is that the dependence of σ disc ( H ) on V is very sensitive, whilethe dependence of σ ess ( H ) on V is far more robust.In any event, let us formulate the first fundamental issue: Fundamental Issue 1.
Study the essential spectrum S = σ ess ( H ).Another consequence of self-adjointness is the existence of spectralmeasures guaranteed by the spectral theorem. Specifically, if H is self-adjoint and ψ ∈ H , then there is a finite Borel measure µ H,ψ on R (indeed, µ H,ψ ( R ) = k ψ k ) such that(2.4) h ψ, ( H − z ) − ψ i = Z R dµ H,ψ ( E ) E − z , z σ ( H ) . In fact, each of the measures µ H,ψ is supported by the spectrum, andhence we could integrate in (2.4) over σ ( H ) rather than all of R . Forthis reason, the structure of the spectrum places restrictions on thespectral measures.For example, it is quite standard to decompose a Borel measure µ on R into its pure point part µ (pp) (which has a countable support),its singular continuous part µ (sc) (which gives no weight to countable D. DAMANIK AND J. FILLMAN sets and has a support of zero Lebesgue measure), and its absolutelycontinuous part µ (ac) (which gives no weight to sets of zero Lebesguemeasure). Thus, our second fundamental issue is the following: Fundamental Issue 2.
Study the spectral measures µ H,ψ and theirLebesgue decomposition.As a specific example of how information regarding the first fun-damental issue may give information regarding the second fundamen-tal issue, imagine that H is such that S can be shown to have zeroLebesgue measure – it then immediately follows that µ (ac) H,ψ vanishes forevery ψ ∈ H .Why would the latter statement be of interest? As alluded to above,the time evolution of interest is given by the unitary group generatedby H . In other words, since the time-dependent Schr¨odinger equationis given by(2.5) i ∂∂t ψ = Hψ, ψ (0) = ψ , self-adjointness of H and the spectral theorem again ensure that wecan write the solution as(2.6) ψ ( t ) = e − itH ψ . The third fundamental issue is therefore the following:
Fundamental Issue 3.
Study the quantum evolution e − itH ψ .There are several results (the RAGE Theorem [14] being the mostprominent) that link the second and third fundamental issue. TheLebesgue decomposition of µ (ac) H,ψ (e.g., information about which of thethree components are non-trivial) gives information about the behaviorof e − itH ψ as | t | → ∞ . As a rather obvious instance of this connection,let us note that by (2.4) and (2.6), the so-called return probability |h ψ , e − itH ψ i| = (cid:12)(cid:12)(cid:12) Z σ ( H ) e − itE dµ H,ψ ( E ) (cid:12)(cid:12)(cid:12) will go to zero as | t | → ∞ if µ H,ψ = µ (ac) H,ψ by the Riemann-LebesgueLemma.To summarize, while we are ultimately interested in addressing thethird fundamental issue, the standard approach to doing so involvesaddressing the other two fundamental issues as well. In this sense, astudy of the (essential) spectrum of H is the most basic of these tasks,and it is the one to which we will devote the remainder of this text. CHR ¨ODINGER OPERATORS WITH THIN SPECTRA 5 The Essential Spectrum S and Things we Might Wantto Know About it As explained in the previous section, we wish to study the essentialspectrum S = σ ess ( H ) of a Schr¨odinger operator H in H . Inspired bythe two-step procedure mentioned earlier, let us begin with the freecase.If V vanishes identically, one can see via the Fourier transform,which conjugates − ∆ to multiplication with | ξ | in L ( R d , dξ ), that σ ( H ) = σ ess ( H ) = S = [0 , ∞ ) since that is the (closure of the) rangeof the multiplication operator in question. Thus, in the free case, S isconnected, that is, it has no gaps.Adding on a potential, one may first wonder whether this propertyis preserved. If the operator V is relatively compact with respect to − ∆, which will hold if the function V decays at infinity in a suitablesense, then the essential spectrum is indeed preserved and continues tobe [0 , ∞ ). If relative compactness fails, one cannot immediately deducethat the essential spectrum is not preserved, but one may confidentlyexpect that this should at least be possible for some such perturbationsof − ∆. Thus, one should expect a positive answer to the followingquestion: Question 1.
Can S have gaps, that is, can it be disconnected?We will describe a mechanism that leads to a positive answer in Sec-tion 4: in one dimension, it suffices to consider (non-constant) periodicpotentials V . These potentials do not decay at infinity but they have asufficiently simple structure that their essential spectra can be studiedquite effectively.Once it is understood that S may have gaps, one may then ask ifone can have arbitrarily many, or even infinitely many, gaps. Thus, werefine the previous question as follows: Question 2.
Can S have infinitely many gaps/infinitely many con-nected components?This question can be answered again in the framework of non-constant periodic potentials, but there is an additional wrinkle: thedimension d now matters a lot! When d = 1, it is quite easy to pro-duce infinitely many gaps, while in the case d > S has infinitely many con-nected components. Must they be intervals or is it ever possible thatthe gaps of S are dense, either in all of R or some portion of it? In other D. DAMANIK AND J. FILLMAN words, can S (or a portion thereof) ever be a (generalized) Cantor setin the sense that it has empty interior? This is the next question wemay ask on our quest towards possible thinness phenomena:
Question 3.
Can the gaps of S be dense? In other words, is it possiblethat S has empty interior?The occurrence of Cantor spectra has been a hot topic since the1980s. To exhibit this phenomenon one has to leave the class of periodicpotentials, since Cantor structures are easily seen to be impossiblein the periodic case. But simply taking the closure of the class ofperiodic potentials in a suitable way does the trick! In other words,potentials that are almost-periodic are good candidates when lookingfor Schr¨odinger operators with Cantor spectra. Much of the existingtheory requires d = 1. This is not surprising as we mentioned abovethat gaps at high energies are atypical for periodic potentials when d > Question 4.
Can the Lebesgue measure of S vanish?While the Cantor set just about anyone is first exposed to is themiddle third Cantor set, which is easily seen to have zero Lebesguemeasure, further explorations then show that there are also “fat Can-tor sets” of positive Lebesgue measure. In the context of Schr¨odingerspectra the earliest examples turned out to have positive Lebesguemeasure, but further, more quantitative, work then was able to give apositive answer to the previous question; see Section 7 for details.Once the Lebesgue measure of S is shown to be zero, this naturallyprompts us to consider fractal dimensions. Specifically, since Lebesguemeasure on R is just the α -dimensional Hausdorff measure for α = 1, itis natural to lower the value of α until S no longer has zero weight withrespect to h α . This transition value if called the Hausdorff dimension of Since all operators that we consider are unbounded, the corresponding spectraare also unbounded and hence never compact. To avoid constantly saying “gener-alized Cantor set”, we will abuse terminology somewhat and say a Cantor set is aclosed set with empty interior and no isolated points.
CHR ¨ODINGER OPERATORS WITH THIN SPECTRA 7 S . There are other ways of studying whether S has fractional dimensionin a suitable sense, for example by considering the (lower or upper) boxcounting (or Minkowski) dimension. If a fractional value occurs forone of these dimensions, it is then of interest to look at the others andsee whether they yield the same value, or whether there are essentialspectra for which the different fractal dimensions indeed differ. Sincethe spectrum is a subset of R , all these fractal dimensions of S will takevalues in [0 , Question 5.
Can S be zero-dimensional?Section 8 explains that this is indeed possible, and this will demon-strate that the existing results have pushed the possible thinness ofessential spectra of Schr¨odinger operators to the extreme, in the senseof topology, measure theory, and fractal geometry. But there is animportant additional aspect in pursuing this quest for extreme results.Only once zero-dimensionality has been established in the case d = 1,can one then address Questions 3–5 in the case d >
1. See Section 8for this discussion.After all of these results, one may wonder whether there is any lowerbound that one can impose on the essential spectrum.
Question 6.
Is there any sense in which the size of the set S can bebounded from below?We will take this up in Section 9. Let us note that if V ( x ) is un-bounded, then it is possible for H to have compact resolvent and hencefor σ ( H ) to consist of isolated eigenvalues of finite multiplicity whichonly accumulate at + ∞ . However, in this case S = ∅ , so we do not viewthis as an interesting counterexample. One may observe this explicitlyfor (say) the quantum harmonic oscillator ( V ( x ) = x ), which canbe explicitly diagonalized in terms of Hermite functions; compare [49,p. 142] or [57, Section 6.4]. Consequently, it is mainly of interest toexhibit the desired spectral results within (say) the class of boundedpotentials, and we shall restrict attention to that case in this note.For this note, we focus primarily on operators as in (2.1), but itwill sometimes be useful to discuss the related discrete Schr¨odingeroperators given by choosing a bounded potential v : Z d → R anddefining h = h v = − ∆ Z d + v in ℓ ( Z d ), where[∆ Z d ψ ]( n ) = X k m − n k =1 ψ ( m ) , D. DAMANIK AND J. FILLMAN and [ vψ ]( n ) = v ( n ) ψ ( n ) for n ∈ Z d .4. Can S have gaps? As discussed, if V ≡
0, the spectrum of H is the half-line [0 , ∞ ),and if V enjoys suitable decay at ±∞ , the essential spectrum remainsundisturbed. Thus, it becomes necessary to work with nonzero po-tentials that do not decay. Of course, if V ( x ) ≡ λ a constant, then σ ( H ) = [ λ , ∞ ), so it is necessary to work with nonconstant potentialsas well. The simplest potentials to work with are those that are peri-odic; we say V : R d → R is periodic if there is a basis { x , . . . , x d } of R d such that V ( x + x j ) ≡ V ( x ) for every 1 ≤ j ≤ d .It is an amazing result from the inverse theory that, in one spacedimension, every non-constant periodic potential has a gap in its spec-trum. Theorem 4.1. If V : R → R is periodic and nonconstant, S has agap. This result is originally due to Borg [8], with a short self-containedproof due to Hochstadt [29]; see also the Historical Remarks and refer-ences in [29] for more.Let us say a little bit more about the determination of the spectrumwhen V is periodic. We emphasize that the approach we discuss here isstrictly one-dimensional; we will discuss the higher-dimensional settinglater. For simplicity, take V to have period one, and let u , u solvethe initial value problem − u ′′ + V u = Eu subject to initial conditions (cid:20) u ′ (0) u ′ (0) u (0) u (0) (cid:21) = (cid:20) (cid:21) . Then, the monodromy matrix Φ( E ) = (cid:20) u ′ (1) u ′ (1) u (1) u (1) (cid:21) and its trace D ( E ) := Tr[Φ( E )] tell essentially the whole story aboutgeneral solutions to(4.1) − u ′′ + V u = Eu, in that one has (cid:20) u ′ ( n ) u ( n ) (cid:21) = Φ( E ) n (cid:20) u ′ (0) u (0) (cid:21) for all n ∈ Z . Thus, asymptotic characteristics of u ( x ) are entirelydictated by behavior of large powers of Φ( E ). In particular, one hasthe following possibilities: CHR ¨ODINGER OPERATORS WITH THIN SPECTRA 9 ED ( E ) D ( E ) = ± a b a b a b = a b Figure 1.
Determination of S from D . • D ( E ) ∈ ( − , or Φ( E ) = ± I . Every solution of (4.1) isuniformly bounded, so E is a generalized eigenvalue of H V , andhence E belongs to the spectrum of H V by Sch’nol’s Theorem[52, 55]. • D ( E ) = ± E ) = ± I . Solutions of (4.1) are lin-early bounded, and E again belongs to the spectrum of H V by Sch’nol’s theorem. • D ( E ) / ∈ [ − , u ± of(4.1) such that u ± decays exponentially at ±∞ and grows ex-ponentially at ∓∞ , and E / ∈ σ ( H ). One can appeal to Sch’nolagain or use u ± to explicitly construct the resolvent of H at E .One can say a bit more about D . Since the spectrum of H V must bereal, any value of E with D ( E ) ∈ [ − ,
2] must be real. In particular, D has all real roots. Moreover, if D ′ ( E ) = 0 for some E ∈ R , onemust have D ( E ) / ∈ ( − , E with D ( E ) ∈ ( − , S is a union of nondegenerate closed bands, S = ∞ [ k =1 [ a k , b k ] , where each band B k = [ a k , b k ] is obtained as the closure of a connectedcomponent of the preimage of ( − ,
2) under D . See Figure 1 for avisualization of D ( E ) and how it determines S .In view of this picture of D , one sees that a gap may only close atenergy E if D ( E ) is tangent to one of the horizontal lines at height ± E . Consequently, it seems like the presence of a closed gapis an unstable phenomenon, and one might expect that a “typical”potential breaks all of these gaps open. We discuss this further in thenext section.Of course, the approach outlined above does not work for d ≥
2, buta closer inspection suggests the correct generalization. Namely, given θ ∈ T := R / Z , we may have D ( E ) = 2 cos(2 πθ ) ∈ [ − ,
2] if and only ifthere is a solution u to (4.1) such that u ( x +1) = e πiθ u ( x ), and it is thisnotion which makes perfect sense in higher dimensions. Namely, if V isperiodic and { x , . . . , x d } is a basis of R d for which V ( x ) ≡ V ( x + x j )for every j , then one seeks solutions of(4.2) − ∆ u + V u = Eu such that(4.3) u ( x + x j ) = e πiθ j u ( x )for some θ = ( θ , . . . , θ d ) ∈ T d := R d / Z d . For every θ , there is acountable set E ( θ ) ≤ E ( θ ) ≤ · · · such that (4.2) enjoys a nontrivialsolution for E = E j ( θ ) satisfying (4.3) (we enumerate the E j ’s accord-ing to multiplicity, which is why we write “ ≤ ”). Then, the k th band isgiven by B k = { E k ( θ ) : θ ∈ T d } , and one has S = [ B k . Each E k can be shown to be non-constant, so each B k is a non-degenerate closed interval. The primary difference between the lower-and higher-dimensional cases is that the bands could only touch in d = 1, but when d ≥
2, the bands may overlap (i.e. one may have b k > a k +1 ). This will be responsible for some of the differences onesees in the higher-dimensional setting, including the added difficulty ofconstructing examples having Cantor spectrum.5. Can S have infinitely many gaps? The answer to this question (in the periodic setting) depends entirelyon the dimension. If d = 1 and, for example, V ( x ) = cos x , then it isknown that S is a disjoint union of infinitely many compact intervals[51, pp. 298–299]. In other words, the spectrum has infinitely manygaps at arbitrarily high energies. This topological structure is knownto be generic among all one-dimensional periodic potentials. For thesake of notation, let C r per ( R ), r ∈ Z + ∪ {∞} denote those V ∈ C r ( R ) Here, and throughout this note, a property that may hold at some point in atopological space is said to be generic if the set on which it holds is a dense G δ . CHR ¨ODINGER OPERATORS WITH THIN SPECTRA 11 such that V ( x + 1) ≡ V ( x ), equipped with the uniform topology for r = 0 and the usual Fr´echet topology when r > Theorem 5.1 (Simon, 1976 [54]) . For generic V ∈ C r per ( R ) , b k < a k +1 for all k . In particular, S has infinitely many open gaps. On the other hand, the picture in higher dimensions is starkly differ-ent. In general, the arguments become quite complex, but one can geta simple idea for how things change in the following simple scenario.First, consider a given V ∈ C ( R ). Denoting the k th band and gapby B k = [ a k , b k ] and G k = ( b k , a k +1 ) respectively, and using | · | todenote the length of an interval, we have: X k (cid:0) π (2 k − − | B k | (cid:1) < ∞ (5.1) X k | G k | < ∞ ;(5.2)see [24, 25]. (In fact, the summands of (5.1) and (5.2) are very eas-ily seen to be bounded by 4 k V k ∞ by general perturbation theory, andthis already suffices for the purposes that we shall undertake presently).Now, let us see what insights this yields in higher dimensions. A poten-tial V : R d → R is called separable if there are potentials W j : R → R for 1 ≤ j ≤ d so that V ( x ) = d X j =1 W j ( x j ) . Writing S = σ ( H V ) and S j = σ ( H W j ), the general theory (see, forinstance, [49, Chapter VIII]) shows that S decomposes as a Minkowskisum:(5.3) S = d X j =1 S j = ( d X j =1 E j : E j ∈ S j ) . Combining the observations (5.1), (5.2), and (5.3) shows the follow-ing: if V : R d → R is a continuous separable periodic potential, then σ ( H V ) has only finitely many gaps. On the basis of this calculation,the following result was quite long-expected: Conjecture 5.2 (Bethe–Sommerfeld Conjecture) . Suppose d ≥ . If V : R d → R is periodic and is such that H V is self-adjoint, then σ ( H V ) has finitely many gaps. This result is quite difficult to prove in the non-separable case, andhas only recently been concluded. An incomplete list of contributions includes [28,32,59–61,66], with the current best available result in [43].Versions for discrete Schr¨odinger operators may be found in [22, 34],with the most general discrete result in [26]. For more on the historyand development of this problem (and the spectral theory of periodicoperators writ large), we recommend Kuchment’s survey [35].6.
Can the gaps of S be dense? From the previous discussion, the answer must be a resounding “no”if one restricts to the case of periodic potentials. Another scenariothat was studied intensively at first was the case of random potentials.Under relatively mild assumptions, it is known that the spectrum of arandom operator is the same for almost every realization and that thisalmost-sure spectrum is given by the closure of the union of the spectraof corresponding periodic realizations; compare [33]. In particular, ifthere are any gaps in the (almost-sure) spectrum, they do not accumu-late. Thus, in the random and periodic cases, the associated operators(almost) always have spectra in which the gaps cannot be dense. Letus note in passing that the study of random operators is itself a deepfield with a vast literature; we refer the reader to [1, 63] and referencestherein.However, in the 1970s and 1980s, interest began to grow in aperiodicpotentials with suitable recurrence properties, motivated by several de-velopments in math and physics. First, the study of electrons subjectedto external magnetic fields (which dates back to the early part of the20th century, compare [27,40,46]) led to interest in quasi-periodic mod-els. Famously, plots of the spectra corresponding to Harper’s model ledto a picture now known as Hofstadter’s butterfly [30], one of the firstfractals observed in physical scenarios. These models led to interest inthe almost-Mathieu operator (AMO), which is a quasi-periodic discreteSchr¨odinger operator h v with(6.1) v ( n ) = v λ,α,ω ( n ) = 2 λ cos(2 π ( nα + ω ))for n ∈ Z , where λ, α, ω ∈ R are called the coupling constant , frequency ,and phase , respectively. We recommend [15, 41] for more about suchquasi-periodic operators, their history, and connections to physics. Ofcourse, if α is rational or λ = 0, the resulting v is periodic and hence canbe handled by a formalism analogous to that described in Section 4, sothe most interesting case is that in which α is irrational and λ = 0. Thesecond major physical development that spurred mathematical study ofaperiodic operators was the discovery of quasicrystals by Shechtman etal. in the early 1980s [53]. We will not give a comprehensive account of CHR ¨ODINGER OPERATORS WITH THIN SPECTRA 13 mathematical quasicrystals, instead referring the reader to [6] and ref-erences therein for background. The central operator-theoretic modelof a quasicrystal is the
Fibonacci Hamiltonian , a discrete Schr¨odingeroperator with potential given by(6.2) v ( n ) = v λ,ω ( n ) = λχ [1 − α, ( nα + ω mod 1) , where λ > ω ∈ R are parameters, again called the couplingconstant and the phase. Here, α is fixed to be ( √ − α , leading to Sturmian potentials).Let us briefly comment on the similarities and differences between(6.1) and (6.2). Both are aperiodic, but both are also “not too far”from being periodic, however in vastly different ways. As an almost-periodic potential, (6.1) exhibits ε -almost periods, that is, nontrivialtranslations t such that k v − v ( · − t ) k ∞ < ε . The Fibonacci potential(6.2) is not almost-periodic, indeed has no nontrivial ε -almost-periodfor any ε ≤ λ . Instead, there are nontrivial translations t , t , . . . and R j → ∞ such that v and v ( · − t j ) agree on [ − R j , R j ]. To summarize,(6.1) enjoys imprecise repetitions at a global scale, while (6.2) enjoysexact local repetitions. These differences inform the different propertiesof these operators, as well as the different tools and techniques thathave been brought to bear in their study.To study an operator with a potential such as (6.1) (or a suitable ana-logue in L ( R )), a natural tactic is to approximate α by rationals, studythe resulting periodic operators, and try to prove suitable approxima-tion results. The problem is that such approximations are inherently local . In particular, these approximations will be close to the originalaperiodic potential on a finite window, and will not at all be good ap-proximations outside that window. To put this into the language ofoperator theory, the resulting periodic approximants approximate theaperiodic operator strongly but not in norm , so one is naturally inter-ested in what operators enjoy a periodic approximation in the normtopology. To wit: a potential V ∈ C ( R d ) is called (uniformly) limit-periodic if it may be uniformly approximated by periodic elements of C ( R d ). For the case of limit-periodic operators, one might expect theirspectral properties to correspond closely to those in the periodic case,but we will see that they can diverge quite far. Let LP( R d ) denote theset of limit-periodic V ∈ C ( R d ), which is a complete metric space inthe uniform metric (note however that it is not closed under sums andhence is not a Banach space). The first result on Cantor spectrum inthe limit-periodic case is due to Moser: Theorem 6.1 (Moser, 1981 [42]) . There exist V ∈ LP( R ) such that S is a Cantor set. Very shortly after Moser’s work, Avron–Simon [5] showed that Can-tor spectrum is generic in LP( R ). Let us also mention related workof Chulaevskii [12, 13] and Pastur–Tkachenko [44, 45]. For more aboutlimit-periodic operators in general, see the survey [16] and referencestherein.Once one understands how the process of periodic approximationworks, one is no longer surprised by theorems showing that Cantorspectrum is generic in the limit-periodic setting. There are a few basicheuristics: First, one can show from the definitions that the spectrumnever has isolated points. Next, one can show that Cantor spectrum isa G δ phenomenon, which is essentially an exercise in point-set topol-ogy and perturbation theory. Concretely, if I is an open interval, theset of V for which I ∩ ρ ( H V ) = ∅ is easily shown to be open. Thus,the intersection of all such sets of potentials as I ranges over intervalshaving rational endpoints is a G δ and is rather clearly the set of poten-tials enjoying nowhere-dense spectrum. Finally, one needs to show thatCantor spectrum is dense in the space LP( R ). This is more technical,but the outline is the following. Start with a periodic potential, andsuccessively produce periodic perturbations of higher and higher peri-ods for which all gaps that can open do open (by using Theorem 5.1),and in such a way that gaps opened in previous steps do not close,taking care that the gaps also cannot close in the limit.Let us observe that the outline suggested in the previous paragraphdoes not work for dimensions d >
1. Concretely, Theorem 5.1 onlyholds for d = 1, and fails spectacularly for d >
1, in view of the Bethe–Sommerfeld conjecture. In particular, the presence of Cantor spectrumin higher dimensions is necessarily more difficult to prove than in the 1Dcase. Nevertheless, we will see in Section 8 that suitable quantitativerefinements of the 1D Cantor results can be used to generate Cantorspectrum in higher dimensions.Since we mentioned the AMO, let us note that it was later shown tohave Cantor spectrum for all λ = 0, all ω ∈ R , and all irrational α [3].Similarly, Cantor spectrum has been established for the Fibonacci (aswell as the more general Sturmian) case for all non-trivial parametervalues [7, 64]. See [15, 41] for further comments on the history andearlier partial results. CHR ¨ODINGER OPERATORS WITH THIN SPECTRA 15 Can the Lebesgue measure of S vanish? As soon as one has a Cantor spectrum, one is interested in the mea-sure properties. In particular, even if the gaps are dense, the spec-trum may still be a “fat” Cantor set and hence have positive measure(or even infinite measure, since S is unbounded). Returning brieflyto the discrete setting, the first two operators known to have zero-measure spectrum of physical interest were the critical AMO ((6.1)with λ = 1) and the Fibonacci Hamiltonian [64], with potential givenby (6.2). On one hand, the zero-measure result for the AMO is verysensitive to the coupling constant, since the measure of the spectrum(for irrational α ) is known to be given by | − | λ || for every λ (seee.g. [4, 31, 36, 37]). In particular, as soon as one perturbs the couplingconstant, the zero-measure property is destroyed, even though the Can-tor property persists. By way of contrast, the Fibonacci Hamiltonianexhibits zero-measure Cantor spectrum for arbitrary λ = 0, and hence,the zero-measure phenomenon is more robust in this scenario.This discussion of stability is significant from the point of view of thedesired results for operators of the form − ∆+ V , which are unbounded.In particular, the high-energy region for H V is analogous to the regimeof small coupling constant in the discrete case. In particular, one shouldnot expect a zero-measure result to hold for an analogue of the AMO in L ( R ), so, in order to generate examples of the desired phenomenon forSchr¨odinger operators in L ( R ), one should look in (say) the Fibonaccisetting, where the desired phenomenon is robust in the small couplingregion.To spare the reader technicalities and notation, let us work in asetting that is simpler than the most general possible setup, but whichstill exhibits the behaviors that we want to discuss. Let u denote the0-1 Fibonacci sequence, given as: u ( n ) = u ( n, ω ) = χ [1 − α, ( nα + ω mod 1)with α = ( √ −
1) as before. Choosing f = f in L ([0 , u by f and each 1 by f and attaching a coupling constant λ ; moreprecisely, take:(7.1) V ω,λ ( n ) = λ X n ∈ Z f u ( n,ω ) ( · − n ) χ [ n,n +1) . Of course, one could simply absorb λ into the f j , but it is of interest to fix f j first and then to observe the scaling behavior as the coupling constant is varied. Theorem 7.1 (D.–F.–Gorodetski, 2014 [17]) . For every f = f , every λ = 0 , and every ω ∈ R , S is a Cantor set of zero Lebesgue measure. The result of [17] is more general than this: one can replace α byan arbitrary irrational number; in fact, one can replace these aperiodic0-1 sequences by any element of a minimal subshift satisfying Bosher-nitzan’s criterion for unique ergodicity [9,10], subject to a suitable localrecognizability criterion on the local potential pieces; see [17] for de-tails. See also [39] for similar results that were proved independently.We specialize to the Fibonacci case because of the physical motivationand because our later results about dimension will require it.Additionally, the zero-measure property also holds for the case oflimit-periodic potentials, and remains stable in the coupling constantfor generic potentials, first shown in the discrete case by Avila [2]. Theorem 7.2 (D.–F.–Lukic, 2017 [20]) . For generic V ∈ LP( R ) , σ ( H λV ) is a zero-measure Cantor set for every λ = 0 . The latter claim is somewhat surprising, since one knows that σ ( H λV ) converges to σ ( H ) = [0 , ∞ ) in the Hausdorff metric as λ → λ ↓
0, the resolvent set of H λV remains a dense, full-measure subset, even though the length of every gap goes to zero atleast as fast as 2 λ k V k ∞ .8. Can S be zero-dimensional? Drawing on ideas from Avila [2], one can refine the Cantor spectrumarguments in a quantitative way. The first step is to go from qualitativebehavior to quantitative behavior in the periodic case. Concretely,we know (from Theorem 5.1) that generic periodic potentials open allpossible gaps in dimension one. One of the key insights of Avila inthe discrete setting was a mechanism that one can use to generate notonly gaps, but to show that the spectrum of an operator of period p can be made exponentially small (i.e. . e − cp for a suitable constant c > Theorem 8.1 (D.–F.–Lukic, 2017 [20], D.–F.–Gorodetski, 2019 [18]) . Within
LP( R ) , the set of V for which dim H ( σ ( H V )) = 0 is dense. CHR ¨ODINGER OPERATORS WITH THIN SPECTRA 17
In fact, for a dense set of V ∈ LP( R ) , one has dim H ( σ ( H λV )) =dim − B ( σ ( H λV )) = 0 for every λ = 0 . The result about Hausdorff dimension is from [20], and the resultabou the lower box-counting dimension is from [18]. The strengthen-ing from Hausdorff to box-counting dimension allows us to concludeanswers to all of our questions about S in higher dimensions. Theorem 8.2 (D.–F.–Gorodetski, 2019 [18]) . There exist V ∈ LP( R d ) for any d ≥ such that S is a Cantor set having zero Lebesgue measure,zero Hausdorff dimension, and zero lower box-counting dimension. The key fact that enables one to make the transition is the followingobservation. If A is a compact set, then(8.1) dim − B d X j =1 A ! ≤ d dim − B ( A ) . In particular, if A has zero lower box-counting dimension, so too does P dj =1 A for any d . Thus, Theorem 8.2 is proved by using Theorem 8.1 toconstruct a 1D potential W for which the associated spectrum has lowerbox-counting dimension zero, constructing a d -dimensional separablepotential of the form V ( x ) = d X j =1 W ( x j ) , and then invoking (8.1) and (5.3). We are sweeping a few niceties aboutfractal dimensions of sums of unbounded sets under the rug here, butthey are not hard to deal with, see, e.g., the appendix to [18].Let us note that one really does need the limit-periodic setting here.One can show that the critical AMO exhibits zero Hausdorff dimensionfor some frequencies [38]; however, we remind the reader that becausethese only hold for λ = 1, one cannot expect to be able to use themto deduce consequences for Schr¨odinger operators in L ( R ). Moreover,one can prove dimensional results for the Fibonacci Hamiltonian, butone cannot zero out its Hausdorff dimension. In particular, one has thefollowing for potentials V ω,λ as in (7.1). Let us denote S λ the essentialspectrum corresponding to V ω,λ (the omission of ω from the notationreflects the fact that S does not depend on ω , which can be seen fromstrong approximation arguments). Theorem 8.3 (D.–F.–Gorodetski, 2014 [17], F.–Mei, 2018 [23]) . Forany f = f in L ([0 , and any λ > , one has dim locH ( S λ , E ) > for every E ∈ S λ . In the small-coupling limit, one has: lim λ → inf E ∈S λ dim locH ( S λ , E ) = 1 . Moreover, for any λ > , in the high-energy limit, we have lim E →∞ inf E ∈S λ ∩ [ E , ∞ ) dim locH ( S λ , E ) = 1 . In particular, the global Hausdorff dimension of S λ is always equal to . The asymptotic statements for the dimension were proved for f ≡ f ≡ f j in [23]. Let us now moveto the higher dimensional setting. We fix f ≡ f ≡ λ , and denoteby S λ the spectrum of H V ω,λ as before. For d ≥
2, write S ( d ) λ = d X j =1 S λ , the d -fold sum (cf. (5.3)). In particular, S ( d ) λ is the spectrum of H ( d ) λ,ω := − ∆ + V ( d ) ω,λ , where V ( d ) ω,λ ( x ) = d X j =1 V ω,λ ( x j ) , x ∈ R d . The dimensional results in Theorem 8.3 suggest (but do not imply!)that one has the following picture for the multidimensional operator H ( d ) ω,λ : the spectrum undergoes a transition from Cantor-like at lowenergies to containing intervals and perhaps eventually a half-line forlarge energies. This was recently proved. Theorem 8.4 (D.–F.–Gorodetski, 2020+ [19]) . Let d ≥ . For any λ > , S ( d ) λ contains a half-line. There exists C ( d ) > such that, for λ ≥ C ( d ) , S ( d ) λ enjoys Cantor structure near the ground state, that is,there is an E such that ( −∞ , E ] ∩ S ( d ) λ is a nonempty Cantor set. When Does it End?
With these increasingly fine results that show that the spectrum canbe made arbitrarily small in the sense of Lebesgue measure and fractaldimensions, one wonders whether there is any hard limit at all to howfar one can push this. In other words, is there some notion of thesize of a set for which one can prove a lower bound on the size of thespectrum, valid for a large class of (bounded) potentials? Of course, aswe already mentioned one does need to insert the word “bounded” here,
CHR ¨ODINGER OPERATORS WITH THIN SPECTRA 19 since there are operators with unbounded potentials having compactresolvent, for which S = ∅ . It turns out that potential theory in thecomplex plane offers a partial answer. Let us emphasize that the resultsin this section only apply to one-dimensional operators. At this time,we are unaware of any similar results for higher-dimensional operators.Let us briefly recall the definitions and refer the reader to [48, 62] fortextbook treatments. For a compact set K ⊆ C , write M ( K ) for theset of all Borel probability measures supported on K . For µ ∈ M ( K ),the (logarithmic) energy of µ is given by E ( µ ) = − Z Z log | x − y | dµ ( x ) dµ ( y ) , with E ( µ ) = + ∞ an allowed value. The (logarithmic) capacity of K isCap( K ) := sup { e −E ( µ ) : µ ∈ M ( K ) } , where e −∞ = 0 by convention. A set with Cap( K ) = 0 is known as a polar set ; such sets play the role of “negligible” sets in potential theory,analogous to the role of measure-zero subsets in measure theory. Polarsets are very small in the senses that we have discussed herein: theyhave Hausdorff dimension zero (hence also Lebesgue measure zero) [11];see also the exercises to [58, Section 3.6]. In terms of these objects, thereis a particularly clean lower bound for discrete Schr¨odinger operatorsin 1D: one must have Cap( S ) ≥
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Department of Mathematics, Rice University, Houston, TX 77005,USA
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