Rank one perturbations and Anderson-type Hamiltonians
aa r X i v : . [ m a t h . F A ] J a n RANK ONE PERTURBATIONS AND ANDERSON-TYPEHAMILTONIANS
CONSTANZE LIAW, , ∗ To the memory of R.G. Douglas. You were not only a vast source of knowledge. Wordscannot fully express my appreciation for your steady advice, your unfaltering support andthe many hours of mathematical discussions we shared.
Abstract.
Motivated by applications of the discrete random Schr¨odinger operator, mathe-matical physicists and analysts, began studying more general Anderson-type Hamiltonians;that is, the family of self-adjoint operators H ω = H + V ω on a separable Hilbert space H , where the perturbation is given by V ω = X n ω n ( · , ϕ n ) ϕ n with a sequence { ϕ n } ⊂ H and independent identically distributed random variables ω n .We show that the the essential parts of Hamiltonians associated to any two realizationsof the random variable are (almost surely) related by a rank one perturbation. This resultconnects one of the least trackable perturbation problem (with almost surely non-compactperturbations) with one where the perturbation is ‘only’ of rank one perturbations. Thelatter presents a basic application of model theory.We also show that the intersection of the essential spectrum with open sets is almostsurely either the empty set, or it has non-zero Lebesgue measure. Introduction
In this spirit, let H be a self-adjoint operator on a separable Hilbert space H . Let { ϕ n } ⊂ H be a sequence of linearly independent unit vectors in H , and let ω = ( ω , ω , . . . ) consistof independent, identically distributed random variables ω n corresponding to a probabilitymeasure on R . Assume that the probability distribution satisfies Kolmogorov’s 0-1 law (seeSubsection 2.5 below).Without going into details about the definition, the Anderson-type Hamiltonian is analmost surely self-adjoint operator associated with(1.1) H ω = H + V ω on H , V ω = X n ω n ( · , ϕ n ) ϕ n . In many applications the vectors ϕ n are mutually orthogonal. However, a priori, the definitionallows the case of non-orthogonal vectors ϕ n . And many of the properties that were originallyproved for mutually orthogonal vectors immediately extend to this case. Mathematics Subject Classification.
Primary 47A55; Secondary 82B44, 81Q10.
Key words and phrases. rank one perturbations, Anderson-type Hamiltonian, Krein–Lifshits spectral shift,discrete random Schr¨odinger operator.The work of C. Liaw was supported by the National Science Foundation under the grant DMS-1802682.
Probably the most important special case of such Anderson-type Hamiltonians is the dis-crete Schr¨odinger operator with random potential on l ( Z d ) given by Hf ( x ) = − △ f ( x ) = − X | n | =1 ( f ( x + n ) − f ( x )) , ϕ n ( x ) = δ n ( x ) = (cid:26) x = n, ω n is distributed according to uniform distribution on the interval [ − c, c ]. Thatjust means that each value in the interval occurs with equal probability. Many Andersonmodels are special cases of an Anderson-type Hamiltonian.From the perspective of classical perturbation theory [13] the main difficulty is that the po-tential V ω is almost surely a non-compact operator, implying that many results from classicalperturbation theory cannot be applied here.On the side we mention an important open problem concerning this perturbation family.The Anderson localization conjecture for weak disorder [2, 8, 15, 16, 28] stands out as onethat has been much attempted. The general question is whether or not an initially localizedwave-packed will spread out over time, or remain localized in space as time moves on. Lit-erature renders a variety of definitions what precisely localization means. For example somedefinitions use the wave operator, while others formulate localization in terms of dynamicalproperties, or the persistence of a non-trivial absolutely continuous part (almost surely). Theconjecture can be formulated with either of these definitions. For simplicity we choose thelatter. To embed the conjecture we mention that for the discrete random Schr¨odinger oper-ator in one dimension, d = 1, operators H ω are known to have trivial absolutely continuousparts (almost surely) whenever c >
0. In higher dimensions, d ≥
2, there is a dimensiondependent threshold c d above which the absolutely continuous parts vanish almost surely,and it is expected that for d ≥ c . Now, it is conjectured thatfor d = 2 the discrete random Schr¨odinger operator has vanishing absolutely continuous part(almost surely) whenever c >
0; no matter how small.In contrast to Anderson-type Hamiltonians stands the seemingly simple problem of per-turbing a self-adjoint operator by an operator of rank one. Namely, for a self-adjoint operator A on H consider the family of self-adjoint rank one perturbations by a vector ϕ ∈ H : A α = A + α ( · , ϕ ) ϕ, α ∈ R . (For details beyond this formal definition, see the discussion surrounding equation (2.2) be-low.)When the underlying Hilbert space H is finite dimensional, we just need to keep track ofthe eigenvalues. However, for infinite dimensional H intricate scenarios can occur that areclosely connected with the boundary values of functions from model spaces.In fact, the problem of rank one perturbations has connections to many interesting topicsin analysis, such as model theory including deBranges–Rovnyak and Sz.-Nagy–Foia¸s modelspaces [7, 17, 19], Nehari interpolation [24], Carleson embeddings [5], singular integral oper-ators [18], and truncated Toeplitz operators [4].With this in mind it becomes clear that, although rank one perturbations are the simplestfrom a perturbation theoretic perspective, their fine properties are extremely rich in nature.While Aronszajn–Donoghue theory captures much of the theory related to rank one pertur-bations, the picture is certainly not complete. For example, we do not know the singularcontinuous spectrum of the perturbed operator A α in terms of properties of the unperturbedoperator A , see e.g. [25].It was surprising when the Simon–Wolff criterion [27] on rank one perturbations was usedto study localization properties of random Jacobi matrices [26]. These ideas were extended ANK ONE PERTURBATIONS AND ANDERSON-TYPE HAMILTONIANS 3 to Anderson-type Hamiltonians and refined [1, 10, 11]. For example, it turns out that undermild conditions, any non-zero vector is cyclic for the Anderson-type Hamiltonian almostsurely.In this manuscript, a new relationship between rank one perturbations and the essentialparts of Anderson-type Hamiltonians is presented. In consideration of the great difference inthe very nature of these two perturbation problems, this seems almost paradoxical. On theone hand this result restricts the spectral behavior of the Anderson-type Hamiltonians. Onthe other hand it shows the great complexity of the problem of rank one perturbations.The proof at hand consists of constructing the spectral measures of the two operators. TheKrein–Lifshits spectral shift function allows us to ensure that the hence constructed operatorsare indeed related by a rank one perturbations. These tools are based on similar observationsmade by A. Poltoratski in [22].1.1.
Outline.
In Section 2 we review related results from perturbation theory. We introduceand remind the reader of a few facts about the Krein–Lifshits spectral shift function for rankone perturbations, and we review on Kolmogorov’s 0-1 law as well as its implications forAnderson-type Hamiltonians.In Section 3 we mention some simple known and some new results: In Subsection 3.1,we provide a short proof for two statements about the deterministic spectral structure ofAnderson-type Hamiltonians. In Subsection 3.2 we focus on the intersection of the essentialspectrum with open sets and show that this intersection is almost surely either the emptyset, or it has non-zero Lebesgue measure, see Theorem 3.2.In Section 4 we state and prove the main result (Theorem 4.1), which roughly says thatthe essential parts of H ω and H η are almost surely with respect to the product measure P × P unitary equivalent modulo a rank one perturbation.2. Preliminaries
Perturbation Theory.
Perturbation theory is concerned with the general question:Given some information about the spectrum of an operator A what can be said about thespectrum of the operator A + B for B in some operator class? Depending on which class ofoperators the perturbation B is taken from we obtain different results of spectral stability,i.e. preservation of parts of the spectrum under such perturbations.Since unitarily equivalent operators (i.e. U AU − = B for some unitary operator U ) areof the same spectral type, we introduce the following notation. We write A ∼ B for twooperators A and B if the operators are unitary equivalent. The notation A ∼ B (mod Class X )is used if there exists a unitary operator U such that U AU − − B is an element of Class X .Here, Class X can be any class of operators, e.g. compact, trace class, or finite rank operators.For self-adjoint operators A and B let us recall the following well-known theorems thatwill be used in the proof of Theorem 3.1 below. Theorem 2.1 (Weyl–von Neumann, see e.g. [13]) . The essential spectra of two self-adjointoperators A and B satisfy σ ess ( A ) = σ ess ( B ) if and only if A ∼ B (mod compact operators ) . Here, the essential part of the spectrum is obtained by removing the isolated eigenvaluesof finite multiplicity from the spectrum.
Theorem 2.2 (Kato–Rosenblum, see e.g. [13]) . If for two self-adjoint operators we have A ∼ B (mod trace class ) then their absolutely continuous parts are equivalent, i.e. A ac ∼ B ac . C. LIAW
We briefly explain how to recover the absolutely continuous part of an operator. First finda spectral measure µ (using the Spectral Theorem with respect to some minimal cyclic set ofvectors) and take its Radon–Nikodym derivative dµdx = dµ ac . The desired part of the operatoris the one that corresponds to this absolutely continuous part of the measure. Remark.
For self-adjoint A and B , Carey–Pincus [6] characterized when two operators arerelated by a rank one perturbation, that is, when we have A ∼ B (mod trace class) . Of course,they must have unitarily equivalent absolutely continuous parts. Outside the continuous spec-trum, they are only allowed discrete parts. And the discrete eigenvalues of A and B (countingmultiplicity) must fall into three categories: (i) those eigenvalues of A with distances from thejoint continuous spectrum having finite l norm (i.e. are trace class), (ii) those eigenvaluesof B with distances from the joint continuous spectrum having finite l norm, and (iii) eigen-values of A and B that can be matched up (via a 1-1 and onto map) so that their differenceshave finite l norm. In the case of purely singular measures (i.e. with trivial absolutely continuous part) the nexttheorem resembles a characterization for A ∼ B (mod rank one). Recall that two operators A and B are said to be completely non-equivalent, if there are no non-trivial closed invariantsubspaces H and H of H such that A | H ∼ A | H . It is not hard to see that two operatorsare completely non-equivalent, if and only if their spectral measures are mutually singular.Here, we mean mutually singular in the sense of measure theory. That is, two measures µ and ν are said to be mutually singular, if there is a measurable set B so that µ ( B ) = 0 and ν ( R \ B ) = 0. Theorem 2.3 (Poltoratski [22]) . Let K ⊂ R be closed. By I = ( x ; y ) , I = ( x ; y ) , . . . denote disjoint open intervals such that K = R \ S I n . Let A and B be two cyclic self-adjointcompletely non-equivalent operators with purely singular spectrum. Suppose σ ( A ) = σ ( B ) = K and assume that for the pure point spectra (consisting of the eigenvalues) of A and B we have σ pp ( A ) ∩ { x , y , x , y , . . . } = σ pp ( B ) ∩ { x , y , x , y , . . . } = ∅ . Then we have A ∼ B (mod rank one ) . The proof of our main result applies the latter theorem as well as Lemma 4.2 below whichallows us to introduce absolutely continuous spectrum (while retaining precise control of thesingular measures).2.2.
Cauchy transform and rank one perturbations.
The deep connection betweenoperator theory and the Cauchy transform Kτ ( z ) = 1 π Z R dτ ( t ) t − z , z ∈ C + , of an operator’s spectral measure τ is well studied. This relationship is frequently usedto learn about the spectral properties of the operator under investigation. The connectionbetween operator theory and the Cauchy transform and the spectral theory of rank oneperturbations is particularly well developed [7, 17, 19, 18, 24]. This connection is one of ourmajor ingredients. Here we merely recall the results that are applied later in this article. ANK ONE PERTURBATIONS AND ANDERSON-TYPE HAMILTONIANS 5
It is well-known that the density/weight function w ∈ L of the absolutely continuous partof the measure can be recovered via dτ ac ( x ) = wdx = lim y ↓ ℑ Kτ ( x + iy ) dx, x ∈ R , (2.1)where ℑ denotes the imaginary part.In Aleksandrov–Clark Theory, the following result plays an essential role. Theorem 2.4. (Poltoratski [21] , also see [12] ). Let τ and ˜ τ be two non-negative measureson the real line such that ˜ τ = f τ + ˜ τ s . Then K ˜ τKτ ( x + iε ) ε → −→ f ( x ) τ s − almost everywhere . Here we always work with measures that satisfy Poisson integrability R dτ ( t ) t +1 < ∞ . Es-pecially when dealing with rank one perturbations, we do often encounter measures with R dτ ( t ) | t | +1 = ∞ . In order to avoid difficulties with convergence it is standard to introduce analternative definition of the Cauchy transform K τ ( z ) = 1 π Z R (cid:18) t − z − tt + 1 (cid:19) dτ ( t ) , z ∈ C + . We use both Kτ and K τ below. Notice that the two behave alike locally, as the integrand − tt +1 is uniformly bounded on R . Although it will not play a role later on it is worthmentioning that (for τ such that Kτ is defined on C + ) the real part of K τ differs from theconjugate Poisson integral by a finite additive constant.The advantage of introducing this alternative definition is that it is possible to define K τ for more general measures τ . Indeed, since t − z − tt +1 behaves like t − as t → ∞ , we canwork with Poisson integrable measures τ and do not need to assume the stronger condition R dτ ( t ) | t | +1 < ∞ .Let A be a self-adjoint (possibly unbounded operator) on a Hilbert space H . Let ϕ be suchthat the corresponding rank one perturbation will be form bounded, i.e. k (1 + | A | ) − / ϕ k H < ∞ ; see [18] and its references for more information. Then we can use quadratic forms todefine the family of rank one perturbations via the formal expression A α = A + α ( · , ϕ ) ϕ, α ∈ R . (2.2)Only focussing on the interesting part of the perturbation problem, we assume that ϕ is acyclic vector for A , i.e. H = span { ( A − z I ) − ϕ : z ∈ C \ R } . To see that we are not restricting generality, notice that on the orthogonal complement ofthe invariant subspace span { ( A − z I ) − ϕ : z ∈ C \ R } for A and A α in H , operator A α isindependent of α .In our setting, it is well-known that ϕ is also a cyclic vector of the operator A α for all α ∈ R . By µ α denote the spectral measure of A α with respect to ϕ . In other words, invokingthe Spectral Theorem, µ α is given by(( A α − z I ) − ϕ, ϕ ) H = Z R dµ α ( t ) t − z for all z ∈ C \ R . We use the notation µ = µ . C. LIAW
With the resolvent formula, it is not difficult to see that the Cauchy transforms of themeasures µ and µ α of the rank one perturbation (2.2) are related via the Aronszajn–Kreinformula Kµ α = Kµ παKµ , (2.3)also see [25, Equation (11.13)].Aronszajn–Donoghue theory (see e.g. [25, Section 12.2]) provides a good picture of thespectrum of the perturbed operator for rank one perturbations. One of its intriguing re-sults says that the singular part of rank one perturbations must move when we change theperturbation parameter α : Theorem 2.5 (Aronszajn–Donoghue) . For coupling constants α = β ∈ R , the singular partsof the corresponding spectral measures µ α and µ β are mutually singular, i.e. ( µ α ) s ⊥ ( µ β ) s . This result was proved by Aronszajn for Sturm–Liouville operators with varying boundaryconditions [3] and by Donoghue in the abstract setting of rank one perturbations [9].Another result within this theory gives a necessary condition for a point to be in theessential support of the singular spectrum of A α . The theorem in this form can easily beextracted from Theorem 6 of [9], which states that the set { x : lim y ↓ Kµ ( x + iy ) = − α − } is a carrier for ( µ α ) s (meaning that ( µ α ) s is trivial outside that set). Theorem 2.6.
We have ( µ α ) s ( { x : lim y ↓ Kµ ( x + iy ) = − α − } ) = 0 . Essential support of the absolutely continuous part of a measure.
In orderto define one of the objects of interest, we isolate the limit supremum from the symmetricdefinition of the Radon-Nikodym derivative.In this spirit, we let τ be a Borel measure on R . Fix ε > x D ε τ ( x ) where D ε τ ( x ) := τ ([ x − ε, x + ε ])2 ε . Note that the denominator equals the Lebesgue measure of interval [ x − ε, x + ε ].The essential support of the absolutely continuous part of a Borel measure τ (on R ) isgiven by ess-supp τ ac = (cid:26) x ∈ R : 0 < lim sup ε → D ε τ ( x ) < ∞ (cid:27) . (2.4) Remark 1.
In order to embed this into classical theory, we mention that the Radon-Nikodymderivative of τ exists at x , if and only if lim sup ε → D ε τ ( x ) = lim inf ε → D ε τ ( x ) < ∞ . Remark 2.
Since the Radon–Nikodym derivative exists almost everywhere (with respect toLebesgue measure) two operators satisfy A ac ∼ B ac if and only if the essential supports ofthe absolutely continuous parts of their spectral measures are equal up to a set of measurezero. Indeed, as described in [25, Section 12.1] , two absolutely continuous measures f ( x ) dx and g ( x ) dx are equivalent if and only of the symmetric difference of the sets { x | f ( x ) = 0 } and { x | g ( x ) = 0 } has Lebesgue measure zero. And the operators that act as multiplicationby the independent variable M x on L ( f ( x ) dx ) and L ( g ( x ) dx ) are unitarily equivalent if andonly if the measures f ( x ) dx and g ( x ) dx are equivalent. It remains to apply Remark 1. ANK ONE PERTURBATIONS AND ANDERSON-TYPE HAMILTONIANS 7
Remark 3.
The same arguments as in Remark 2 also imply that the essential support of theabsolutely continuous part of an operator’s spectral measure is up to a set of measure zeroindependent of the choice of cyclic vector (used in the spectral theorem).
It is worth presenting a simple example to demonstrate that ess-supp τ ac ( supp τ ac mayhappen: Example.
Let τ be the measure given by the sum of Lebesgue measures on intervals that haveall rational points of [0 , as centers and with width − n +1 . Namely, with an enumeration { q n } of these rational points, let dτ ( x ) = X n ∈ N χ [ q n − − n ,q n +2 − n ] ( x ) dx. The sum of the interval width P n ∈ N − n +1 = 2 , so that the Lebesgue measure of the essentialsupport satisfies the crude estimate | ess-supp ac τ | ≤ . On the other hand, the rationals aredense in [0 , and so ≤ | supp τ ac | . In fact, as and are centers of some intervals, wehave < | supp τ ac | . In any case, we have ess-supp τ ac ( supp τ ac . Krein–Lifshits Spectral Shift for Rank One Perturbations.
In this section, webriefly present the Krein–Lifshits spectral shift function and its properties for rank one per-turbations. More detailed explanations, examples and proofs can be found in [23] and thereferences therein.Consider the rank one perturbations A α given by (2.2) and their spectral measures µ α corresponding to the cyclic vector ϕ .Since the spectral measure µ is non-negative, the Cauchy transform Kµ ( z ) is Herglotz,i.e. its imaginary part is non-negative for z ∈ C + . For every α ∈ R it is hence possible to findan essentially bounded by − π < u ( t ) ≤ π , t ∈ R , function and a constant c ∈ R such that1 + παKµ = e K u + c . (2.5)see e.g. [20, Section VIII.1]. To better understand this formula, recall that the angular bound-ary values of the Cauchy transform exist almost everywhere with respect to the Lebesguemeasure. Now think of K u as the analytic upper half-plane extension of u . So that for α > A and A α so that α > u can equivalently be defined viathe principal argument u = arg(1 + παKµ ) . (2.6)Function u is called the Krein–Lifshits spectral shift of the rank one perturbation A α . Since Kµ is Herglotz, the range of u is contained in [0 , π ] . Indeed, consider the logarithm of (2.5),take its imaginary part and recall the relation (2.1).By breaking Kµ in (2.6) into real and imaginary part Kµ = iP µ − Qµ (where P denotesthe Poisson integral and Q denotes the conjugate Poisson integral), it becomes clear that thesingularity of the integrand causes u to jump from 0 to π at isolated points of supp µ s .In the non-isolated case, a characterization of the point masses of µ and µ α is included in[20, Section VIII.5].Using the Aronszajn–Krein formula (2.3) we obtain a relation between the shift functionand the measure µ α : 1 − παKµ α = e − K u − c . And the analog u = − arg(1 − παKµ α )(2.7) C. LIAW of (2.6) for µ α implies that u drops from π to 0 at isolated points of supp( µ α ) s .So in essence, each family of spectral measures { µ α } α ∈ R corresponds to some Krein–Lifshitsspectral shift function u .Further the set where u ∈ (0 , π ) and not equal to one of the endpoints of the inverval isequal (up to a set of Lebesgue measure zero) to ess-supp( µ ) ac . In particular, it follows thatess-supp( µ ) ac = ess-supp( µ α ) ac . Remark 4.
These observations about the relationship between the spectrum of A and A α ,and the behavior of u give an alternative proof for the fact that the discrete spectrum of twopurely singular operators in the same family of rank one perturbations must be interlacing.In absence of absolutely continuous spectrum, u can only take on the values and π , so thatthe Krein–Lifshits spectral shift essentially jumps from to π and then back. Vice versa, it is well-known that for fixed α > u which is essen-tially bounded by 0 ≤ u ≤ π is the Krein–Lifshits spectral shift of the rank one perturbation M µ + α ( · , ) of the multiplication operator M µ by the independent variable on L ( µ ). Infact, given such a function u and α > µ and ν = µ α ifwe impose a normalization condition on the measures. For α = 1 we say that the measures µ and ν correspond to u .2.5. Kolmogorov’s 0-1 law and Anderson-type Hamiltonians.
Consider triples (Ω , A , P )of probability spaces, where Ω = R ∞ consists of countably many copies of R and where P isa countable product of equal probability measures. We let ω = ( ω , ω , . . . ) ∈ Ω be taken inaccordance with P .Here we consider only those probability measures P that satisfy Kolmogorov’s 0-1 law.Namely, properties that are invariant under changing finitely many of the ω n are enjoyedwith probability 0 or 1. This is particularly useful here, because perturbation theory tells usthat many properties are independent under finite rank perturbations.Specifically, we use: Proposition 2.7 (Kolmogorov’s 0-1 law applied to Anderson-type Hamiltonians) . Considerthe Anderson-type Hamiltonian H ω given by (1.1) . Assume that the probability distribution P satisfies the 0-1 law. Then those spectral properties that are invariant under finite rankperturbations are enjoyed by H ω almost surely or almost never. Deterministic spectral structure
Deterministic absolutely continuous part and essential spectrum.Theorem 3.1.
Let H ω be given by (1.1) . Assume the hypotheses of Section 1 and assume that P satisfies the Kolmogorov 0-1 law. Then almost surely with respect to the product measure P × P :1) ( H ω ) ac ∼ ( H η ) ac and H ω ∼ H η (mod compact operator ) . While the statement in item 1) is known (see [10, Corollary 1.3]), we present a short prooffor the convenience of the reader and since the proof structure also underlies the proof of thestatement in item 2).
Proof.
The words ‘almost surely’ (‘almost never’) in this proof refer to almost surely (almostnever) with respect to the product measure P × P . ANK ONE PERTURBATIONS AND ANDERSON-TYPE HAMILTONIANS 9
Let H e ω denote finite rank perturbations of H , i.e. e ω = ( e ω , e ω , . . . ) with e ω n = 0 only forfinitely many n . In particular, H e ω are compact and trace class perturbations of H .To show the statement in item 1). Without loss of generality, let µ ω denote the ‘fiber’ ofthe spectral measure of H ω for which ess-supp µ ω is maximal with respect to the inclusionof sets. (Alternatively, one can think of µ ω as the associated scalar-valued spectral measure.This can also be obtained by taking the trace of a matrix-valued spectral measure.) Let µ e ω be the analog measure for H e ω .By the Kato–Rosenblum theorem (see Theorem 2.2) and Remarks 1 and 2, for almost every x ∈ R we have x ∈ ess-supp( µ (0 , , ,... ) ) ac if and only if x ∈ ess-supp( µ e ω ) ac . By virtue of theKolmogorov 0-1 law (see Proposition 2.7), for almost every x ∈ R we have x ∈ ess-supp( µ ω ) ac almost surely or almost never. The set (up to a set of measure zero) of points x for whichthe latter is almost surely true is hence deterministic and the statement in item 1) is proven.Item 2) follows in analogy via the Weyl–von Neumann theorem (see Theorem 2.1) replacingTheorem 2.2. (cid:3) Remark 5. (a) In fact, we have proved the stronger – than item 1) of Theorem 3.1 –statement that the essential support of the absolutely continuous spectrum is a deterministicset (up to a set of Lebesgue measure zero). Namely, for some measurable set A ⊂ R we havethat the symmetric difference A △ ess-supp( µ ω ) ac has Lebesgue measure zero P almost surely ω .(b) Similarly for item 2) of Theorem 3.1, it follows that there exists a deterministic set K such that K = σ ess ( H ω ) almost surely.(c) Although the perturbation V ω is almost surely (with respect to P ) a non-compact pertur-bation, there is still a deterministic set K = σ ess ( H ω ) for P almost all ω . Intersection of the essential spectrum with open sets.
Assume the setting ofTheorem 3.1. Recall that σ ess ( H ω ) is a deterministic set, by item 2) of Theorem 3.1. Theorem 3.2.
Assume the hypotheses of Theorem 3.1 and assume that P is a product ofabsolutely continuous measures. Let O be an open set and let X = O ∩ σ ess ( H ω ) . Then almostsurely either X = ∅ , or the Lebesgue measure | X | > . Proof.
Assume | X | = 0 and X = ∅ . Take x ∈ X .Since O is open, there exists ε > x − ε, x + ε ) ⊂ O . Consider X ε = X ∩ ( x − ε, x + ε ). Clearly we have | X ε | = 0.Recall item 1) of Theorem 3.1. This implies that almost surely( µ ω ) ac (( x − ε, x + ε )) = ( µ ω ) ac ( X ε ) = 0 . In virtue of Lemma 3.3 below ( µ ω ) s ( X ε ) = 0 almost surely.Therefore x / ∈ σ ess ( H ω ) almost surely, in contradiction to the fact that x ∈ X . Hencealmost surely either X = ∅ or | X | > (cid:3) Lemma 3.3.
Assume the hypotheses of Theorem 3.1 and assume that P is a product ofabsolutely continuous measures µ k . If set A ⊂ R satisfies | A | = 0 , then we have ( µ ω ) s ( A ) = 0 almost surely.Proof. Recall that P is a product of absolutely continuous measures µ k .Assume that ( µ ω ) s ( A ) > k ∈ N ) thereexist ω and X ⊂ R such that µ k ( X ) > α ∈ X we have ( µ ω α ) s ( A ) > ω α = ω + αδ k . But this contradicts the Aronszajn–Donoghue Theorem 2.5 for rank one perturbations.Notice that X contains at least two points, since all µ k are absolutely continuous. (cid:3) Almost sure unitary equivalence modulo a rank one perturbation
The main result of this paper, see Theorem 4.1 below, says that the essential parts of twoAnderson-type Hamiltonians are unitarily equivalent modulo a rank one perturbation. Itsproof relies on constructing an appropriate Krein–Lifshits spectral shift function.By ∂S we denote the boundary of a given set S , and by | · | denote the Lebesgue measure. Theorem 4.1.
Assume the hypotheses of Theorem 3.1. Assume that ( H ω ) ess is cyclic almostsurely (with respect to P ) and P = Π k µ k is a product measure of purely absolutely continuousmeasures µ ω on R . Let µ denote the spectral measure of the operator ( H ω ) ess with respect tosome cyclic vector. If | ∂ ess-supp( µ ω ) ac | = 0 almost surely, then ( H ω ) ess ∼ ( H η ) ess (mod rank one ) almost surely with respect to the product measure P × P . On the one hand, this result greatly restricts the possible deterministic properties ofAnderson-type Hamiltonians. On the other hand, it tells us how ‘wild’ rank one pertur-bations can be.Recall that the essential spectrum comes about from removing from the spectrum all iso-lated point masses that have finite multipilicity. Further recall that the absolutely continuousand singular parts of the spectrum arise from Lebesgue decomposition of its spectral mea-sure, µ = µ ac + µ s . A particular decomposition of the operator is then obtained throughunitary equivalence with the particular decomposition of the spectral representation. (Thatis, on the spectral representation side, the L ( µ ) space is orthogonally decomposed in accor-dance with the particular spectral decomposition, the multiplication operator is restricted tothese invariant subspaces, and the decomposition of the operator is carried over via unitaryequivalence.) Remark 6. (a) If a family of Anderson-type Hamiltonians possesses a weak Anderson local-ization property (namely, if there is no absolutely continuous spectrum almost surely), thenthe hypotheses of cyclicity and | ∂ ess-supp( µ ω ) ac | = 0 hold automatically. Indeed, the re-stricted operator ( H ω ) s is cyclic almost surely by Theorem 1.2 of [11] , and also recall thatthe operators ( H ω ) ac and ( H ω ) s are completely non-equivalent because the essential supportsof their spectral measures are mutually singular. Similarly, almost sure cyclicity of ( H ω ) ac implies the almost sure cyclicity of ( H ω ) ess .(b) In the conclusion of this result it is necessary to restrict to the essential parts of theoperators. The statement H ω ∼ H η (mod rank one ) is not true, since the finite isolated pointspectra of H ω and H η might not interlace. This intertwining is one of the necessary conditionsfor two operators to be unitarily equivalent up to rank one perturbation. In fact, between twopoints in the discrete spectrum of H ω there may be any number of points from the discretespectrum of H η (almost surely).(c) Theorem 4.1 cannot be concluded trivially by using Theorem 2.3, plus item 1) of Theorem3.1 and then separating the singular from the absolutely continuous part. This can be seenby counterexample: Embedded singular spectrum can occur for one operator, but not for theother (with positive probability). In particular, the absolutely continuous spectrum of ( H ω ) ess may have dense embedded singular spectrum, and ( H η ) ess has purely absolutely continuousspectrum. In this case, the singular parts of ( H ω ) ess and ( H η ) ess are not unitarily equivalentup to rank one perturbations (as they would have to interlace). ANK ONE PERTURBATIONS AND ANDERSON-TYPE HAMILTONIANS 11 (d) We expect that relaxing the hypotheses of the theorem from ( H ω ) ess is cyclic to assumingthat it has finite multiplicity m would yield the conclusion ( H ω ) ess ∼ ( H η ) ess (mod rank m ) . The proof of Theorem 4.1 uses Poltoratski’s result on a characterization of rank one per-turbations in terms of the spectrum (Theorem 2.3) as well as the following lemma which willallow us to introduce absolutely continuous spectrum while retaining precise control of thesingular measures.
Lemma 4.2.
Let u be a Krein–Lifshits spectral shift function with range in the set { , π } . Let µ and ν be the corresponding spectral measures. Take an open set O ⊂ R such that | O | < ∞ .For c > define a new shift function by ˜ u ( x ) = (cid:26) u ( x ) on R \ O | u ( x ) − min { dist( R \ O, x ) , π/ }| , if x ∈ O. For the measures ˜ µ and ˜ ν that correspond to ˜ u , we have the equivalence of measures ˜ µ | R \ O ∼ µ | R \ O and ˜ ν | R \ O ∼ ν | R \ O .Proof. For t ∈ R \ O we have | K ( u − ˜ u )( t ) | ≤ Z O (cid:12)(cid:12)(cid:12)(cid:12) u ( x ) − ˜ u ( x ) t − x (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ Z O dist( R \ O, x ) | t − x | dx ≤ | O | , and with (2.5), it follows that0 < c < πK ˜ µ πKµ < C < ∞ µ | R \ O − almost everywhere . (Since ˜ µ and ˜ ν correspond to ˜ u , we have by convention α = 1.)By definition µ | R \ O and ˜ µ | R \ O are purely singular. Therefore, we have0 < ˜ c < K ˜ µKµ < ˜ C < ∞ µ | R \ O − almost everywhere . (4.1)If (on R \ O ) measure µ has a part that is singular with respect to ˜ µ (denote it by η ),then the ratio of Cauchy integrals K ˜ µKµ tends to zero with respect to η almost everywhere.This contradicts the lower bound of the last estimate (4.1). Hence µ | R \ O must be absolutelycontinuous with respect to ˜ µ | R \ O .The other direction – that ˜ µ | R \ O is absolutely continuous with respect to µ | R \ O – followsin analogy and we have proven ˜ µ | R \ O ∼ µ | R \ O . The result for ν can be proven in analogy. (cid:3) Proof of Theorem 4.1.
Most of this proof is to be understood almost surely with respect tothe product measure P × P , although this might not be stated everywhere explicitly.By µ denote the spectral measure of the operator ( H ω ) ess with respect to some cyclic vectorand similarly for ν and ( H η ) ess , where ( ω, η ) is distributed according to P × P . It is worthmentioning that the spectral measures of an operator corresponding to any two cyclic vectorsare equivalent.In virtue of Lemma 4.3 (below) we have that µ s ⊥ ν s almost surely with respect to productmeasure.The goal is to produce a spectral shift function with corresponding spectral measures thatare equivalent to the spectral measures µ and ν , respectively. This is done by construction of auxiliary measures µ and ν that behave like µ and ν on the singular parts. And in a secondstep we modify these auxiliary measures to obtain the desired absolutely continuous parts.In the end, we verify that we did not destroy the good singular behavior that the auxiliarymeasures had.By item 1) of Theorem 3.1, the symmetric differenceess-supp µ ac △ ess-supp ν ac is a set of measure zero (almost surely with respect to the product measure). Let us denote theintersection of these sets by F = ess-supp µ ac ∩ ess-supp ν ac . Notice that by the hypothesis,without loss of generality, we can assume | ∂ ess-supp µ ac | = | ∂ ess-supp ν ac | = 0. A simple settheoretic argument shows that | ∂F | = 0.Further, by item 2) of Theorem 3.1 and the Weyl–von Neumann theorem, Theorem 2.1,their essential spectra satisfy σ ess ( H ω ) = supp µ = supp ν . Let us denote this set by E = σ ess ( H ω ) . First observe that, by definition of E , operators ( H ω ) ess and ( H η ) ess have dense purelysingular spectrum on the set E \ clos( F ). By the definition of F and since | ∂F | = 0, it ispossible to choose two purely singular measures µ ′ and ν ′ such that: • µ ′ and ν ′ are mutually singular ( µ ′ ⊥ ν ′ ), • µ ′ | R \ ( F \ ∂F ) = ν ′ | R \ ( F \ ∂F ) = 0, and so that • µ = µ s + µ ′ and ν = ν s + ν ′ have dense (alternating) spectrum on E .The rough idea is that µ | R \ ( F \ ∂F ) and ν | R \ ( F \ ∂F ) are essentially what we are lookingfor. Further, µ and ν are spectral measures of operators that are rank one perturbations ofone another. We still need to modify these measures on F \ ∂F , in order to ensure that theconstructed measures are equivalent to µ and ν also on F .By Theorem 2.3, the measures µ and ν possess a spectral shift function u , i.e. thereexists a function u which is essentially bounded by 0 ≤ u ≤ π and such that u = arg(1 + πKµ ) = − arg(1 − πKν ) . Note that the hypothesis that there are no point masses at the endpoints is satisfied almostsurely. So we can assume this condition without loss of generality.In order to destroy the artificially created singular spectrum and introduce the appropriateabsolutely continuous spectrum, we define u ( x ) = (cid:26) u ( x ) , if x ∈ R \ ( F \ ∂F ) , | u ( x ) − min { dist( R \ ( F \ ∂F ) , x ) , π/ }| , if x ∈ F \ ∂F, and let µ and ν be the measures corresponding to u .It remains to prove that µ ∼ µ and ν ∼ ν . We will explain the equivalence of µ and µ .The same fact for ν follows in analogy.Let us begin with the absolutely continuous parts. Recall that | ∂F | = 0. So on the set F we have u ∈ (0 , π ) Lebesgue almost everywhere. By equations (2.6), (2.7) and (2.1), itfollows that dµ dx ( x ) > < ∞ for Lebesgue almost all x ∈ F . This means that( µ ) ac | F ∼ ( µ ) ac | F . The equivalence of the absolutely continuous part on R \ F follows similarly from the factthat u takes only the values 0 or π on R \ F .We have shown that ( µ ) ac ∼ µ ac . And by the same reasoning we have ( ν ) ac ∼ ν ac . ANK ONE PERTURBATIONS AND ANDERSON-TYPE HAMILTONIANS 13
Now we need to ensure that this construction lead to the desired singular parts. By thedefinition the measures we ensured that on the complement of the interior of F (on the set R \ ( F \ ∂F )) we have the equality of measures µ | R \ ( F \ ∂F ) = ( µ ) s | R \ ( F \ ∂F ) = µ | R \ ( F \ ∂F ) and Lemma 4.2 implies µ | R \ ( F \ ∂F ) ∼ ( µ ) s | R \ ( F \ ∂F ) ∼ µ | R \ ( F \ ∂F ) . It remains to check the singular parts on F \ ∂F . We begin by recalling that in definition(2.4) the points where the limit-superior is infinite are excluded. So by the definition of F via the intersection of essential supports of the absolutely continuous measures we have that µ s | F \ ∂F ≡
0. By the definition of u on F \ ∂F , the same is true for ( µ ) s . Indeed, for anyclosed set X ⊂ F \ ∂F there exists an ε > u ( x ) ∈ ( ε, π − ε ) for all x ∈ X . Byequation (2.7), this means thatlim y ↓ ℑ Kν ( x + iy ) = 0 for all x ∈ X. In virtue of Theorem 2.6 (applied to the measures µ α = µ and µ = ν ) it follows that( µ ) s ( X ) = 0. Whereby the singular parts satisfy the desired property also on F \ ∂F . (cid:3) If the { ϕ n } form an orthonormal sequence, the following lemma is proved as a corollary tothe main theorem in [10]. Although, their proof extends immediately to the non-orthogonalcase, we decided to include a new shorter proof here. Lemma 4.3.
Assume the hypotheses of Theorem 3.1 and assume that P is a product ofabsolutely continuous measures. Then ( µ ω ) s ⊥ ( µ η ) s almost surely with respect to the productmeasure. In particular (with the notation of the proof of Theorem 4.1), we have µ s ⊥ ν s almost surely with respect to the product measure.Proof. Assume that the set S = { ( ω, η ) : ( µ ω ) s ( µ η ) s } has positive product measure.Because P is assumed to be a product of absolutely continuous measures, there then exists apair ( ω, η ) ∈ S such that H ω is a rank one perturbation of H η . But by Aronszajn–Donoghuetheory, see Theorem 2.5, this is not possible. (cid:3) Acknowledgments.
The author would like to thank Alexei Poltoratski for suggesting theproblems which led to this paper as well as for the many insightful discussions and commentsalong the way. Further thanks to the referees.
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Conference report(for the conference held in April 2009 at the Banff International Research Station). Department of Mathematical Sciences, University of Delaware, 501 Ewing Hall, Newark,DE 19716, USA;CASPER, Baylor University, One Bear Place
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