Algebraic localization implies exponential localization in non-periodic insulators
aa r X i v : . [ m a t h - ph ] J a n ALGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION INNON-PERIODIC INSULATORS
JIANFENG LU AND KEVIN D. STUBBS
Abstract.
Exponentially-localized Wannier functions are a basis of the Fermi projection of aHamiltonian consisting of functions which decay exponentially quickly in space. In two and threespatial dimensions, it is well understood for periodic insulators that exponentially-localized Wannierfunctions exist if and only if there exists an orthonormal basis for the Fermi projection with finitesecond moment (i.e. all basis elements satisfy R | x | | w ( x ) | d x < ∞ ). In this work, we establish asimilar result for non-periodic insulators in two spatial dimensions. In particular, we prove that ifthere exists an orthonormal basis for the Fermi projection which satisfies R | x | ǫ | w ( x ) | d x < ∞ for some ǫ > Introduction
In electronic structure theory, we are often interested in localized representations of the occupiedspace. For band-insulating materials (i.e., materials where the Fermi energy lies the in the band gap)these local representations can be studied by constructing a well localized basis for a subspace knownas the Fermi projection. Over the past few decades, exponentially localized Wannier functions(ELWFs) have emerged as the basis of choice for the Fermi projection in periodic insulators. Amongthe many applications, Wannier functions have been instrumental in constructing effective tightbinding models and in the development of the modern theory of polarization. We refer the readers tothe review article [12] and references therein for more detailed discussions on the Wannier functionsfrom a physical point of view.Given the importance of ELWFs, there have been much research devoted to understanding whenELWFs exist. As a result of these efforts, it is now well understood that for periodic insulators indimension one, two, and three the existence of ELWFs is tied to the vanishing of certain topologicalinvariants. We give a quick review of these theoretical results • In one spatial dimension, a basis of ELWFs for Fermi projection of an insulating crystallinematerial always exists [2, 4, 15, 6, 16]. • In two dimensions, the same result holds if and only if the Chern number, a topologicalinvariant associated with the Fermi projector, vanishes [1, 18, 13]. • In three dimensions, the result holds as long as three “Chern-like” numbers all vanish[1, 18, 13].To complement these results which connect ELWFs and topology, in [13] the authors were able toshow that for periodic systems in two dimensions that the Chern number vanishes if and only if thereexists a basis of Wannier functions with finite second moment (and similarly in three dimensions).This result, combined with the previous results connecting ELWFs to the Chern number, formsthe basis of the localization-topology correspondence or
Localization Dichotomy . Informally stated,
Date : January 11, 2021.This work is supported in part by the U.S. National Science Foundation via grant DMS-2012286 and the U.S. De-partment of Energy via grant DE-SC0019449. K.D.S. was supported in part by a National Science FoundationGraduate Research Fellowship under Grant No. DGE-1644868. this correspondence is the following result in two and three dimensions: ∃ an orthonormal basis, { w α } , for range ( P ) s.t. for all α , Z R | x | | w α ( x ) | d x < ∞⇐⇒ The Chern number vanishes / The “Chern-like” numbers vanish ⇐⇒ ∃ γ ∗ > , ∃ an orthonormal basis, { ˜ w α } , for range ( P ) s.t.,for all α , Z R e γ ∗ | x | | ˜ w α ( x ) | d x < ∞ Far less is known for systems which are not periodic however it has been conjectured that aLocalization Dichotomy should also hold in non-periodic systems [10]. The main result of thepaper is to establish a weaker Localization Dichotomy for both periodic and non-periodic systems.In particular, we show that that one can establish the equivalence between exponential and algebraiclocalization without connecting to the theory of topological invariants. Using the informal notationintroduced above, the main result of this paper is to show in two dimensions that: ∃ an orthonormal basis, { w α } , for range ( P ) s.t. for some ǫ > α , Z R | x | ǫ | w α ( x ) | d x < ∞ , ⇐⇒ ∃ γ ∗ > , ∃ an orthonormal basis, { ˜ w α } , for range ( P ) s.t.,for all α , Z R e γ ∗ | x | | ˜ w α ( x ) | d x < ∞ This result lends support to the Localization Dichotomy Conjecture for non-periodic systems re-cently proposed by Marcelli, Monaco, Moscolari, and Panati in [10, 11]. The precise statement ofthese results follows in the next section.1.1.
Technical Statement of Results.
We begin by specifying our assumptions on the Hamil-tonian H : Assumption 1 (Regularity on H ) . Throughout this paper, we will assume that the Hamiltonian H takes the following form H = ( i ∇ + A ) + V where A ∈ L ∞ ( R ; R ) , div( A ) ∈ L ∞ ( R ; R ) , and V ∈ L ∞ ( R ; R ) . Remark 1.1.
Following the analysis given in [11], we expect that our regularity assumptions canbe relaxed to A ∈ L ( R ; R ), div( A ) ∈ L ( R ), and V ∈ L ( R ). We assume that A , div( A ),and V are all L ∞ here so that we can directly appeal to some of the results from our previous work[20] where no attempt was made to optimize the regularity assumptions. For more details on therole of these assumptions see Remark 5.4.Using standard techniques it is easy to verify that Assumption 1 implies H is essentially self-adjoint on L ( R ) [19]. Our second assumption on H concerns its spectrum: Assumption 2 ( H has a spectral island) . We suppose that H has a spectral island . That is wecan decompose the spectrum of H as σ ( H ) = σ ∪ σ where dist ( σ , σ ) > and diam ( σ ) < ∞ . To state the main results of this paper, we will now make two important definitions: finitelydegenerate centers and generalized Wannier basis . Our definition for generalized Wannier basisdiffers slightly from those given previously in [10, 11]. We will discuss these differences in moredetail in Section 1.2.
Definition 1 (Finitely Degenerate Centers) . We say that a collection of functions { ψ α } α ∈I ⊆ L ( R ) has finitely degenerate centers if: LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 3 (1) Each ψ α has a well defined expected position. That is, for all α ∈ I the following quantitiesexist and are finite:(1) µ Xα := h ψ α , Xψ α i and µ Yα := h ψ α , Y ψ α i . We will refer to the point ( µ Xα , µ Yα ) as the center point for ψ α .(2) The collection of center points for { ψ α } α ∈I has bounded density. That is, there exists aconstant M < ∞ such that for all ( x , y ) ∈ R we have |{ α : ( µ Xα , µ Yα ) ∈ B ( x , y ) }| ≤ M where B ( x , y ) is a ball of radius 1 centered at ( x , y ). Definition 2 (Generalized Wannier Basis) . Let P ∈ B ( L ( R )) be an orthogonal projection. Wesay that a collection of functions { ψ α } α ∈I is a generalized Wannier basis if there exists alocalization function G ( · ) : R → R + and a constant C > { ψ α } α ∈I forms an orthonormal basis for range ( P ) with finitely degeneratecenters.(2) Each ψ α is localized about its center point in the sense that Z R G ( | x − ( µ Xα , µ Yα ) | ) | ψ α ( x ) | ≤ C where ( µ Xα , µ Yα ) is as defined in Equation (1).In this paper, we specifically consider two important special classes of generalized Wannier bases:(1) We say a generalized Wannier bases is exponentially localized if we can choose G ( | x | ) = e γ | x | for some γ > s -localized if we can choose G ( | x | ) = (1 + | x | ) s forsome s > Theorem 1 (Main Theorem) . Let P be the spectral projector onto σ for a Hamiltonian H satis-fying Assumptions 1 and 2. Then the following statements are equivalent:(1) P admits a generalized Wannier basis that is exponentially localized.(2) P admits a generalized Wannier basis that is s -localized for some s > / . Connection with Previous Work and Discussions.
Generalized Wannier bases in twodimensions have been considered and defined in previous works [14, 10, 11], however the definitionwe give in our work slightly differs from these previous definitions. The main difference betweenour definition of a generalized Wannier basis and the ones given in these previous works lies in theconditions imposed on the center points. In our work, we define the center point of a basis function ψ α to be its expected position in X and Y . While this requires that each basis element has a welldefined expected position, this assumption is satisfied since we assume that P admits an s -localizedbasis for s > /
2. We then additionally assume that these center points do not cluster arbitrarilystrongly to prevent pathological counterexamples.In the work by Nenciu-Nenciu [14], the set of center points were only assumed to be a discreteset. We found this assumption to be problematic as it allows the center points to become arbitrarilystrongly clustered. In the work of Marcelli, Monaco, Moscolari, and Panati [10], the authors assumethat the center points are part of a Delone set (a set which is “nowhere dense” and “nowheresparse”, see [10, Definition 5.2]). This Delone set assumption puts a fairly rigid structure on thecenter points which does not play a role in our argument since we only focus on the localization ofthe Wannier functions. The definition of a generalized Wannier basis given in the more recent workby Marcelli, Moscolari and Panati [11], on the other hand, is equivalent to the finitely degeneratecenters assumption so long as the basis has a well defined expected position.
JIANFENG LU AND KEVIN D. STUBBS
Before continuing with the remainder of the paper we will quickly discuss the Localization Di-chotomy Conjecture introduced in [10]. In the periodic case, the Chern number plays an importantrole in classifying when an exponentially localized basis of Wannier functions exists or not. Thenatural extension of these ideas to the non-periodic case was introduced in [10] (see also [3]) and isknown as
Chern marker . Definition 3 (Chern Marker) . Let P be a projection on L ( R ) and χ L be the indicator functionof the set ( − L, L ] . The Chern marker of P is defined by C ( P ) := lim L →∞ πi L tr (cid:16) χ L P h [ X, P ] , [ Y, P ] i P χ L (cid:17) whenever the limit on the right hand side exists.With this definition the Localization Dichotomy Conjecture as stated in [10] is the following: Conjecture 1 (Localization Dichotomy Conjecture) . Let P be the spectral projector onto σ for aHamiltonian H satisfying Assumptions 1 and 2. Then the following statements are equivalent:(a) P admits a generalized Wannier basis that is exponentially localized.(b) P admits a generalized Wannier basis that is s -localized for s = 1 .(c) P is topologically trivial in the sense that its Chern marker C ( P ) exists and is equal to zero. The main result of this work shows that (b) ⇒ (a) for s > /
2. In recent work [11], Marcelli,Moscolari, and Panati have shown that (b) ⇒ (c) for s >
5. Since an exponentially localized basisis also s -localized for any s ≥
0, our result combined with the result from [11] implies that (b) ⇒ (c) for s > / Notation and Conventions
We begin by fixing some notations. Vectors in R d will be denoted by bold face with theircomponents denoted by subscripts. For example, v = ( v , v , v , · · · , v d ) ∈ R d . For any v ∈ R d , weuse | · | to denote its Euclidean norm. That is | v | := (cid:18) d X i =1 v i (cid:19) / For any f : R → C , we will use k f k to denote the L -norm of f defined as follows: k f k := (cid:18)Z R | f ( x ) | d x (cid:19) / . Similarly, for any linear operator we will use k A k to denote the induced norm when we view A asa mapping L ( R ) → L ( R ). That is, k A k := sup f ∈ L ( R ) f =0 k Af kk f k . Given two sets
A, B ⊆ R we define their diameter and distance as follows:diam ( A ) := sup {| a − a | : a , a ∈ A } dist ( A, B ) := inf {| a − b | : a ∈ A, b ∈ B } For any contour in the complex plane, C , we will use ℓ ( C ) to denote the length of C . LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 5
Exponentially Tilted Operators.
Given a point ( a, b ) ∈ R , and a non-negative constant γ ≥
0, we define an exponential growth operator, B γ, ( a,b ) , by B γ, ( a,b ) := exp (cid:16) γ p X − a ) + ( Y − b ) (cid:17) = e γ | ( X − a, Y − b, | . where X and Y are the standard position operators are defined as follows: Xf ( x ) = x f ( x ) Y f ( x ) = x f ( x ) . Given a linear operator A , we define(2) A γ, ( a,b ) := B γ, ( a,b ) AB − γ, ( a,b ) . We refer to A γ, ( a,b ) as “exponentially-tilted” relative to A . We will often prove estimates wherewe use the notation (2) but omit the point ( a, b ). In this case the estimate should be understoodas uniform in the choice of point ( a, b ). As a note, per our convention, when γ = 0, A γ, ( a,b ) = A .Throughout this paper, we will assume that the Fermi projector P is fixed to be the spectralprojector onto σ for H satisfying Assumptions 1 and 2. For this projection we also define theoperators Q , P γ , Q γ : Q := I − PP γ := B γ, ( a,b ) P B − γ, ( a,b ) Q γ := B γ, ( a,b ) QB − γ, ( a,b ) = I − P γ where we have used our convention for exponentially tilted operators defined above. Observe that P γ = (cid:16) B γ, ( a,b ) P B − γ, ( a,b ) (cid:17)(cid:16) B γ, ( a,b ) P B − γ, ( a,b ) (cid:17) = B γ, ( a,b ) P B − γ, ( a,b ) = P γ so P γ is also a projection. Similarly, it can be checked that Q γ is also a projection.3. Proof Sketch
The main idea of our proof can be traced back to a proposition by Kivelson for defining Wannierfunctions in non-periodic system. In [7], Kivelson proposed defining generalized Wannier func-tions to be the eigenfunctions of the projected position operator
P XP . To support this proposal,Kivelson showed that the exponentially localized Wannier functions found by Kohn in [8] are infact eigenfunctions of
P XP . After the work by Kivelson, Niu [17] argued heuristically that in onedimension the eigenfunctions of
P XP should decay faster than any polynomial. Later, Nenciu-Nenciu [14] showed rigorously under very general assumptions in one dimension the operator
P XP has discrete spectrum and its eigenfunctions are exponentially localized.In recent work [20] (see also [21]), we were able to generalize the proof of Nenciu-Nenciu [14] totwo dimensions by identifying a sufficient condition for the existence of an exponentially localizedbasis for range ( P ). Specifically, we introduced the assumption of “uniform spectral gaps” on theoperator P XP : Assumption 3 (Uniform Spectral Gaps) . Suppose that there exist constants ( d, D ) such that:(1) There exists a countable set, J , such that: σ ( P XP ) = [ j ∈J σ j . (2) The distance between σ j , σ k ( j = k ) is uniformly bounded from below: d := min j = k (cid:16) dist( σ j , σ k ) (cid:17) > . JIANFENG LU AND KEVIN D. STUBBS (3) The diameter of each σ j is uniformly bounded: D := max j ∈J (cid:16) diam( σ j ) (cid:17) < ∞ . Assuming
P XP has uniform spectral gaps allows one to show constructively that range ( P ) hasan exponentially localized basis. We summarize the idea of the argument in [20] in the followingfew paragraphs.Under Assumptions 1 and 2 on the Hamiltonian H , it can be shown that P XP is an essentiallyself-adjoint operator. Hence, if we assume that
P XP has uniform spectral gaps we can define the band projectors for
P XP via the spectral theorem as follows:
Definition 4 (Band Projectors for
P XP ) . Suppose that
P XP is essentially self-adjoint and sat-isfies the uniform spectral gaps assumption for a collection of sets { σ j } j ∈J . We define the bandprojectors for P XP as the collection of orthogonal projectors { P j } j ∈J where P j is the spectralprojection associated with σ j .Intuitively speaking, since P j is a spectral projection for P XP associated with the bounded set σ j ,we can expect that functions from range ( P j ) will be concentrated in the strip { ( x, y ) ∈ R : x ∈ σ j } .Additionally, since there is some separation between the “bands” of P XP , using techniques fromCombes-Thomas-Agmon theory, it can be shown that the projectors P j are exponentially localizedin a sense to be specified below.Since functions from range ( P j ) are concentrated along a strip, when we restrict our focus torange ( P j ) we have essentially reduced our two dimensional system to a one dimensional system.Therefore, since P j is exponentially localized, it is possible to extend the argument given by Nenciu-Nenciu [14] to show that the eigenfunctions of P j Y P j are exponentially localized. In [20], we provethat for Hamiltonians satisfying Assumptions 1 and 2 if P XP has uniform spectral gaps then:(1) The operator P j Y P j is essentially self-adjoint.(2) The operator P j Y P j has discrete spectrum.(3) The eigenfunctions of P j Y P j are exponentially localized in both X and Y directions simul-taneously.Recalling our convention for exponentially tilted operators (Section 2.1), the technical statementproven in [20] is the following: Theorem 2 (from [20, Section 7]) . Suppose that P is the spectral projector onto σ for a Hamil-tonian satisfying Assumptions 1 and 2. Suppose further that there exists a collection of orthogonalprojectors { P j } j ∈J and finite, positive constants ( γ ∗ , K ′′ , K ′′ , K ′′ ) such that for all ≤ γ ≤ γ ∗ (1) The projectors { P j } j ∈J decompose range ( P ) into orthogonal subspaces in the sense that(a) P j ∈J P j = P .(b) P j P k = ( P j , j = k ;0 , j = k. (2) Each P j is exponentially localized in the sense that(a) k P j,γ − P j k ≤ K ′′ γ ,(b) k [ P j,γ , Y ] k ≤ K ′′ .(3) Each P j is concentrated along a line of the form x = ξ j in the sense that for each P j thereexists a ξ j ∈ R such that: k ( X − ξ j ) P j,γ k ≤ K ′′ and k P j,γ ( X − ξ j ) k ≤ K ′′ . Then the following is true for each j ∈ J : LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 7 (1) The operator P j Y P j is essentially self-adjoint.(2) The operator P j Y P j has discrete spectrum.(3) There exists finite, positive constants ( C, ˜ γ ) , independent of j , such that if ψ ∈ range ( P j ) and P j Y P j ψ = ηψ , then Z e γ √ x − ξ j ) +( y − η ) | ψ ( x, y ) | d x d y ≤ C. Therefore by the spectral theorem, the collection of all eigenfunctions of the operators { P j Y P j } j ∈J form a complete, exponentially localized, orthogonal basis for range ( P ) . An important point of Theorem 2 is that it only makes assumptions about a collection of or-thogonal projections { P j } j ∈J . These orthogonal projections could be spectral projectors of theoperator P XP but this is not strictly necessary. The main result of this paper is to show that ifthere exists an generalized Wannier basis for range ( P ) which is s -localized for s > /
2, then it ispossible to construct an essentially self-adjoint operator b X such that:(1) b X is close to the true position operator X .(2) b X is exponentially localized.(3) The operator P b XP has uniform spectral gaps.Using these properties, it can be shown that the spectral projectors for P b XP satisfy the assumptionsof Theorem 2. Therefore, if we take { P j } j ∈J to be the band projectors for P b XP , then Theorem 2implies that diagonalizing P j Y P j for each j ∈ J gives an exponentially localized orthogonal basisfor range ( P ). 4. Construction of b X Throughout the rest of this paper, we will assume that P admits an s -localized generalizedWannier basis with s > /
2. That is, there exists an orthonormal basis { ψ α } α ∈I for range ( P ) suchthat:(1) { ψ α } α ∈I has finitely degenerate centers;(2) { ψ α } α ∈I is s -localized for some s > / b X will be to observe some important consequences of the finitelydegenerate centers assumption.4.1. Consequences of Finitely Degenerate Centers.
For each ( m, n ) ∈ Z let us define theunit box centered ( m, n ) as follows: S m,n := (cid:20) m − , m + 12 (cid:19) × (cid:20) n − , n + 12 (cid:19) . If a basis { ψ α } satisfies the finitely degenerate centers assumption we know that there is some finitenumber M such that no more than M basis elements have their center in S m,n . Using this property,we can relabel our basis as { ψ ( j ) m,n } ( m,n ) ∈ Z where ψ ( j ) m,n has its center in S m,n and j is a degeneracyindex which runs from 1 to M . In the case that the box S m,n has fewer than M basis elements(say it has j ∗ ) then we define ψ ( j ) m,n ≡ j > j ∗ .The importance of this relabelling is that it allows us to essentially discretize the center pointsof the basis { ψ ( j ) m,n } . Recall that if ψ ( j ) m,n is s -localized then by definition(3) Z (1 + | x − µ Xm,n,j | + | x − µ Ym,n,j | ) s | ψ ( j ) m,n ( x ) | d x ≤ C. JIANFENG LU AND KEVIN D. STUBBS where ( µ Xm,n,j , µ
Ym,n,j ) is the center point of ψ ( j ) m,n . Furthermore, by definition we know that:(4) | µ Xm,n,j − m | ≤
12 and | µ Ym,n,j − n | ≤ µ Xm,n , µ
Ym,n ) and ( m, n ) are close, we have the following lemma:
Lemma 4.1. If ψ ( j ) m,n satisfies Equation (3) for some s ≥ , then it also satisfies (5) Z (1 + | x − m | + | x − n | ) s | ψ ( j ) m,n ( x ) | d x ≤ s ( C + 1) Hence, treating the center point of ψ ( j ) m,n as ( m, n ) instead of ( µ Xm,n , µ
Ym,n ) only makes the boundsworse by a constant factor.Proof. Observe that for any ( x, y ) ∈ R and ( m, n ) ∈ R we have1 + | x − m | + | x − n | = | ( x − m, x − n, | = | ( x − µ Xm,n,j , x − µ Ym,n,j , − ( µ Xm,n,j − m, µ Ym,n,j − n, | (6)Now recall the elementary inequality for any a, b ∈ R d (which is an immediate consequence oftriangle inequality): | a + b | s ≤ s ( | a | s + | b | s ) . Therefore, using this inequality in (6) we have(1 + | x − m | + | x − n | ) s ≤ s (cid:16) | ( x − µ Xm,n,j , x − µ Ym,n,j , | s + | ( µ Xm,n,j − m, µ Ym,n,j − n, | s (cid:17) Since | µ Xm,n,j − m | ≤ and | µ Ym,n,j − n | ≤ it’s clear that for any s ≥ | ( µ Xm,n,j − m, µ Ym,n,j − n, | ≤
1. Therefore, we conclude that: Z (1+ | x − m | + | x − n | ) s | ψ ( j ) m,n ( x ) | d x ≤ Z s (cid:18) (1 + | x − µ Xm,n,j | + | x − µ Ym,n,j | ) s + 1 (cid:19) | ψ ( j ) m,n ( x ) | d x ≤ s ( C + 1) . (cid:3) Another important consequence of knowing an s -localized basis with finitely degenerate centersfor range ( P ) exists is any other localized basis must also have finitely degenerate centers. Lemma 4.2.
Suppose that P ∈ B ( L ( R )) is an orthogonal projector. Suppose further that { ψ α } and { φ α } are two distinct orthonormal bases for range ( P ) and both bases are s -localized with s > .If { ψ α } has finitely degenerate centers then { φ α } must also have finitely degenerate centers.Proof. Given in Appendix A. (cid:3)
Because Lemma 4.2 and our assumptions, once we construct an exponentially localized basis forrange ( P ) we know that this basis must necessarily have finitely degenerate centers. LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 9
The Construction of b X . Since we have discretized the center points of the basis { ψ ( j ) m,n } ,we can use this discretization to define a new position operator ˜ X :(7) ˜ X := X m,n,j m | ψ ( j ) m,n ih ψ ( j ) m,n | + QXQ.
Since
P Q = QP = 0 it’s clear that σ ( P ˜ XP ) ⊆ Z . Furthermore, if we assume that the basis { ψ ( j ) m,n } has sufficiently fast algebraic decay then it’s possible to show that k ˜ X − X k = O (1). Unfortunately,since the basis { ψ ( j ) m,n } only decays algebraically, the band projectors for P ˜ XP will generally not beexponentially localized. To address this issue, we use the spectral filter approach used by Hastingsin [5]. Specifically, we define b X via the formula:(8) b X := Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ ˜ Xe − i ( Xt + Y t ) / ∆ d t d t Here ∆ is a finite parameter to be chosen as part of our proofs and f ( t ) is a filter function definedin terms of its Fourier transform as follows:ˆ f ( ξ ) = ( (1 − | ξ | ) | ξ | ≤ | ξ | ≥ f ( ξ ) is C ( R ), tf ( t ) ∈ L ( R ). Also, note that R f ( t ) d t = ˆ f (0) = 1.To better understand how this construction gives exponential localization for the band projectors,let’s consider the simple case where ˜ X and b X are both finite matrices. In this case, for any λ = ( λ x , λ y ) and µ = ( µ x , µ y ), we can find vectors v λ and v µ which are simultaneous eigenvectorsof X and Y so that Xv λ = λ x v λ Y v λ = λ y v λ Xv µ = µ x v µ Y v µ = µ y v µ For these vectors we have that h v λ , b Xv µ i = Z R f ( t ) f ( t ) h v λ , e i ( Xt + Y t ) / ∆ ˜ Xe − i ( Xt + Y t ) / ∆ v µ i d t d t = h v λ , ˜ Xv µ i Z R f ( t ) f ( t ) e i ( λ x t + λ y t ) / ∆ e − i ( µ x t + µ y t ) / ∆ d t d t = h v λ , ˜ Xv µ i (cid:18)Z R f ( t ) e i ( λ x − µ x ) t / ∆ d t (cid:19) (cid:18)Z R f ( t ) e i ( λ y − µ y ) t / ∆ d t (cid:19) = h v λ , ˜ Xv µ i ˆ f (cid:18) λ x − µ x ∆ (cid:19) ˆ f (cid:18) λ y − µ y ∆ (cid:19) (9)Since ˆ f is only non-zero on ( − , | λ x − µ x | ≥ ∆ or | λ y − µ y | ≥ ∆ then h v λ , b Xv µ i = 0. This calculation shows that in this simple case that the formula Equation (8) setsthe entries of ˜ X far from the diagonal to zero (hence b X is a local operator). Unfortunately, thiscalculation does not seem to generalize to the continuum case due to the fact we do not have goodcontrol on the oscillation of functions from range ( Q ). Despite this technical difficulty, we can stillshow that b X satisfies some technical estimates needed to prove the estimates in Theorem 2.4.3. Proof Organization.
The remainder of this paper is devoted to proving properties of b X andthe operator P b XP and is organized as follows. We begin by stating a number of decay estimatesrequired on the the Fermi projector P in Section 5. In Section 6 we show that b X as constructedabove is close to the true position operator X . In Section 7 we will prove some technical estimateswhich imply band projectors of P b XP are exponentially localized. Finally, in Section 8 we will show that P b XP has uniform spectral gaps so the band projectors for P b XP exist and are well defined.Having proved these important estimates for b X , we will finish the proof of Theorem 1 in Section 9by showing that the band projectors of P b XP satisfy the assumptions of Theorem 2.5. Decay Estimates on P A key part underlying our proofs in the next few sections are some operator norm estimateson the projector P . At a high level, these estimates quantify the statement “ P is exponentiallylocalized” in a technical sense. The most fundamental of these estimates is the following bound.There exists a C > γ sufficiently small k P γ k ≤ C. Similar to the calculation in Equation (9), this bound is best understood when the system isfinite. As before, for any λ = ( λ x , λ y ) and µ = ( µ x , µ y ), we can find vectors v λ and v µ which aresimultaneous eigenvectors of X and Y so that Xv λ = λ x v λ Y v λ = λ y v λ Xv µ = µ x v µ Y v µ = µ y v µ By the definition of the spectral norm, we have that k P γ k ≤ C = ⇒ |h v µ , B γ, ( a,b ) P B − γ, ( a,b ) v λ i| ≤ C = ⇒ e γ | ( µ x − a,µ y − b, | e − γ | ( λ x − a,λ y − b, | |h v µ , P v λ i| ≤ C = ⇒ |h v µ , P v λ i| ≤ Ce − γ ( | ( µ x − a,µ y − b, |−| ( λ x − a,λ y − b, | ) = ⇒ |h v µ , P v λ i| ≤ C ′ e − γ √ µ x − λ x ) +( µ y − λ y ) where in the last line we have set ( a, b ) = ( λ x , λ y ). From this calculation, we see that in the positionbasis, the off diagonal entries of P decay exponentially quickly.Formally, our proof requires the following technical estimates which we verify are true for anyHamiltonian satisfying Assumptions 1 and 2 (see Lemma 5.2): Assumption 4 (Decay Estimates for P ) . We say that the orthogonal projector P satisfies decayestimates , if there exist finite, positive constants ( γ ∗ , C , C , C , C , C ) such that for all ≤ γ ≤ γ ∗ and any λ ∈ R we have the following operator norm bounds:(i) k P γ − P k ≤ C γ (ii) (a) k [ P, X ] k ≤ C (b) k [ P, Y ] k ≤ C (iii) (a) k [ P γ − P, X ] k ≤ C γ (b) k [ P γ − P, Y ] k ≤ C γ (iv) k [ P, h X − λ i / ] k ≤ C (v) (a) kh X − λ i / [ P, X ] h X − λ i − / k ≤ C .(b) kh X − λ i / [ P, Y ] h X − λ i − / k ≤ C . Assuming that P satisfies the bounds in Assumption 4, it is straightforward to verify the followingadditional bounds (which we make use of in our proofs) also hold: Corollary 5.1. If P satisfies decay estimates (Assumption 4), then P also satisfies the followingestimates for all γ ≥ sufficiently small (including γ = 0 ): LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 11 (i) k P γ k , k Q γ k = O (1) (ii) k P γ XQ γ k , k Q γ XP γ k , k P γ Y Q γ k , k Q γ Y P γ k = O (1) (iii) k [ P γ , X ] k , k [ P γ , Y ] k = O (1) (iv) k P γ XQ γ − P XQ k , k Q γ XP γ − QXP k = O ( γ ) (v) kh X − λ i / P Y Q h X − λ i − / k , kh X − λ i / P XQ h X − λ i − / k = O (1) (vi) kh X − λ i / QY P h X − λ i − / k , kh X − λ i / QXP h X − λ i − / k = O (1) Proof.
We will show that Assumption 4 implies kh X − λ i / P Y Q h X − λ i − / k = O (1) . The remaining parts of the corollary follow by similar techniques, and will be omitted. We calculate h X − λ i / P Y Q h X − λ i − / = h X − λ i / P [ Y, Q ] h X − λ i − / = h X − λ i / P h X − λ i − / h X − λ i / [ Y, Q ] h X − λ i − / = −h X − λ i / P h X − λ i − / h X − λ i / [ Y, P ] h X − λ i − / Hence kh X − λ i / P Y Q h X − λ i − / k≤ kh X − λ i / P h X − λ i − / kkh X − λ i / [ Y, P ] h X − λ i − / k≤ k [ h X − λ i / , P ] h X − λ i − / + P kkh X − λ i / [ Y, P ] h X − λ i − / k≤ (cid:16) k [ h X − λ i / , P ] k + 1 (cid:17) kh X − λ i / [ Y, P ] h X − λ i − / k which is bounded due to Assumption 4(iv,v). (cid:3) Importantly, the spectral projectors for Hamiltonians H satisfying Assumptions 1 and 2 alsosatisfy our decay estimates: Lemma 5.2.
Suppose that H is a Hamiltonian satisfying Assumptions 1 and 2. If P is a spectralprojector onto σ , then P satisfies decay estimates (Assumption 4).Proof. See Appendix B. (cid:3)
Remark 5.3.
If one can establish the projector P has an exponentially decaying kernel in thesense that there exists finite, positive constants ( C, γ ) such that: | P ( x , x ′ ) | ≤ Ce − γ | x − x ′ | then it is easy to verify Assumption 4 is true (cf. [11]). Remark 5.4.
The main proofs in this work only rely on the specific properties of the Hamiltonian H in two places(1) Establishing that P satisfies decay estimates as in Assumption 4.(2) A compactness result (in particular, our previous work [20] requires that the operator f ( H + i ) − is compact for any f ∈ C ∞ c ( R )).If one can establish these two facts, then all the results of the paper follow. In particular, sinceoperator norm estimates are still well defined in discrete systems, our results can be easily extendedto L ( Z ) and as well as finite systems. Closeness of b X and X The main goal of this section is to prove the following proposition.
Proposition 6.1.
Suppose that P is a projector satisfying decay estimates as Assumption 4. Sup-pose further that P admits a basis with finitely degenerate centers which is s -localized for some s > , then there exists a finite constant C > such that k b X − X k ≤ C. Let’s start the proof of this proposition with a straightforward calculation. By definition of b X we have that: b X − X = Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ ˜ Xe − i ( Xt + Y t ) / ∆ d t d t − X = Z R f ( t ) f ( t ) (cid:16) e i ( Xt + Y t ) / ∆ ˜ Xe − i ( Xt + Y t ) / ∆ − X (cid:17) d t d t = Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ ( ˜ X − X ) e − i ( Xt + Y t ) / ∆ d t d t (10)where we have used that R f ( t ) d t = 1 and the fact that [ X, e − i ( Xt + Y t ) ] = 0. Therefore, k b X − X k ≤ Z R k f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ ( ˜ X − X ) e − i ( Xt + Y t ) / ∆ k d t d t ≤ k ˜ X − X k (cid:18)Z R | f ( t ) | d t (cid:19) (cid:18)Z R | f ( t ) | d t (cid:19) Since f ∈ L ( R ), the proposition is proved so long as we can show that k ˜ X − X k is bounded. Let’srecall the definition of ˜ X (7, revisited) ˜ X = X m,n,j m | ψ ( j ) m,n ih ψ ( j ) m,n | + QXQ.
Now since P + Q = I we have that X − ˜ X = ( P + Q ) X ( P + Q ) − ˜ X = P XP + P XQ + QXP + QXQ − ˜ X = (cid:16) P XP − X m,n,j m | ψ ( j ) m,n ih ψ ( j ) m,n | (cid:17) + (cid:16) P XQ + QXP (cid:17) . (11)Now at least formally we can write: P XP = X m,n,j | ψ ( j ) m,n ih ψ ( j ) m,n | X X m ′ ,n ′ ,j ′ | ψ ( j ′ ) m ′ ,n ′ ih ψ ( j ′ ) m ′ ,n ′ | = X m,n,j X m ′ ,n ′ ,j ′ h ψ ( j ) m,n , Xψ ( j ′ ) m ′ ,n ′ i| ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ | . Since { ψ ( j ) m,n } is an orthonormal basis, we have that when ( m, n, j ) = ( m ′ , n ′ , j ′ ): h ψ ( j ) m,n , Xψ ( j ′ ) m ′ ,n ′ i = h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i . Therefore, we can express the difference from Equation (11) as follows:
P XP − X m,n,j m | ψ ( j ) m,n ih ψ ( j ) m,n | LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 13 = X m,n,j X m ′ ,n ′ ,j ′ h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ | . Hence, k ˜ X − X k ≤ k X m,n,j X m ′ ,n ′ ,j ′ h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ |k + k P XQ k + k QXP k Since k P XQ k + k QXP k is bounded by Corollary 5.1(ii) with γ = 0, to finish the proof we onlyneed to control the operator norm of the first term, which also justifies the expansion above. Todo this, we will appeal the following proposition which we prove in Appendix D. This propositionis little stronger than we need to prove Proposition 6.1, but we will need to use this stronger resultas part of the proofs in Section 7. Proposition 6.2.
Fix an orthonormal basis { ψ ( j ) m,n } . For any h, g ∈ L ( R ) define h m,n,j and g m ′ ,n ′ ,j ′ as follows: h m,n,j := Z R | ψ ( j ) m,n ( x ) h ( x ) | d x g m ′ ,n ′ ,j ′ := Z R | ψ ( j ′ ) m ′ ,n ′ ( x ) g ( x ) | d x . If { ψ ( j ) m,n } is an s -localized basis with s > then there exists an absolute constant C > such that X m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| h m,n,j g m ′ ,n ′ ,j ′ ≤ C k h kk g k Noting that h m,n,j = h| h | , | ψ ( j ) m,n |i and g m ′ ,n ′ ,j ′ = h| ψ ( j ′ ) m ′ ,n ′ | , | g |i . This result proves Proposition 6.1since by definition k X m,n,j X m ′ ,n ′ ,j ′ h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ |k = sup k h k , k g k =1 | X m,n,j X m ′ ,n ′ ,j ′ h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ ih h, ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ , g i|≤ sup k h k , k g k =1 X m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i|h| h | , | ψ ( j ) m,n |ih| ψ ( j ′ ) m ′ ,n ′ | , | g |i . b X is exponentially localized The main goal of this section is to prove the following proposition which gives a precise meaningto the statement “ b X is exponentially localized”. Proposition 7.1.
Suppose that P is a projector satisfying decay estimates as Assumption 4. Sup-pose further that P admits a basis with finitely degenerate centers which is s -localized for some s > , then there exist finite constants C , C > such that for any µ ∈ R and any γ sufficientlysmall: k b X γ − b X k ≤ C γ (12) k ( X − µ + i ) − [ b X γ , Y ]( X − µ + i ) − k ≤ C (13)In the discrete case, proving Equation (12) implies that the matrix entries of b X decay exponen-tially quickly off the diagonal. Therefore, this bound captures at least part of the statement “ b X is exponentially localized”. On the other hand, Equation (13) controls the non-commutativity of b X and Y . The need for this bound arises as a technical condition needed to prove that the bandprojectors of P b XP satisfy the assumptions of Theorem 2 (in particular it is used to prove Equation (33b)). Despite this difference in interpretation, the techniques to prove these two bounds are quitesimilar. We will prove Equation (12) in Section 7.1 and Equation (13) in Section 7.27.1. Proof of Equation (12) . As we saw in Equation (10) in Section 6, using the fact that R R f = 1and [ X, e − i ( Xt + Y t ) / ∆ ] = 0 we have that: b X − X = Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ ( ˜ X − X ) e − i ( Xt + Y t ) / ∆ d t d t . Hence, since ˜ X = P ˜ XP + QXQ and X = P XP + QXQ + P XQ + QXP we can rewrite ˜ X − X in the integrand above and obtain: b X − X = Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ ( P ˜ XP − P XP − QXP − P XQ ) e − i ( Xt + Y t ) / ∆ d t = Z R f ( t ) f ( t ) (cid:16) e i ( Xt + Y t ) / ∆ ( P ˜ XP − P XP ) e − i ( Xt + Y t ) / ∆ − e i ( Xt + Y t ) / ∆ ( QXP + P XQ ) e − i ( Xt + Y t ) / ∆ (cid:17) d t d t . To reduce clutter in the next few steps, let’s define the following shorthands: A (1) := Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ ( P ˜ XP − P XP ) e − i ( Xt + Y t ) / ∆ d t d t A (2) := Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ ( QXP + P XQ ) e − i ( Xt + Y t ) / ∆ d t d t Using this notation we clearly have that(14) b X − X = A (1) − A (2) . Multiplying on the left by B γ and on the right by B − γ we have that(15) b X γ − X = A (1) γ − A (2) γ , where we have made use of our convention for exponentially tilted operators (Section 2.1). Usingthe identities in Equations (14) and (15) we can rewrite the difference we are interested in boundingas follows: b X γ − b X = ( b X γ − X ) − ( b X − X )= ( A (1) γ − A (2) γ ) − ( A (1) − A (2) )= ( A (1) γ − A (1) ) − ( A (2) γ − A (2) )Hence to show that k b X γ − b X k ≤ C γ , it is enough to find constants K , K so that k A (1) γ − A (1) k ≤ K γ k A (2) γ − A (2) k ≤ K γ We will show the bound for A (1) in Section 7.1.1 and the bound for A (2) in Section 7.1.2. LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 15
Bounding k A (1) γ − A (1) k . To begin, we will first write
P XP − P ˜ XP as an integral kernel.Repeating the calculations from Section 6, we see that we can write the difference P XP − P ˜ XP as follows: P XP − P ˜ XP = X m,n,j X m ′ ,n ′ ,j ′ h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ | Therefore, we can define the integral kernel K : R × R → R as follows:(16) K ( x , y ) = − X m,n,j X m ′ ,n ′ ,j ′ h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i ψ ( j ) m,n ( x ) ψ ( j ′ ) m ′ ,n ′ ( y ) . Using this kernel, for any g ∈ C ∞ c ( R ) we have that:(( P ˜ XP − P XP ) g )( x ) = Z R K ( x , y ) g ( y ) d y . We can then use this kernel to express the action of A (1) on any arbitrary g ∈ C ∞ c ( R ):( A (1) g )( x ) = Z R f ( t ) f ( t ) e i ( x t + x t ) / ∆ (cid:18)Z R K ( x , y ) e − i ( y t + y t ) / ∆ g ( y ) (cid:19) d t d t = Z R K ( x , y ) g ( y ) (cid:18)Z R f ( t ) e i ( x − y ) t / ∆ d t (cid:19) (cid:18)Z R f ( t ) e i ( x − y ) t / ∆ d t (cid:19) d y = Z R K ( x , y ) g ( y ) ˆ f (cid:18) x − y ∆ (cid:19) ˆ f (cid:18) x − y ∆ (cid:19) d y Slightly abusing notation we define(17) ˆ f (cid:18) x − y ∆ (cid:19) := ˆ f (cid:18) x − y ∆ (cid:19) ˆ f (cid:18) x − y ∆ (cid:19) . With this notation we have( A (1) g )( x ) = Z R K ( x , y ) ˆ f (cid:18) x − y ∆ (cid:19) g ( y ) d y. Now recall our definition for B γ : B γ = B γ, ( a,b ) = exp (cid:16) γ p X − a ) + ( Y − b ) (cid:17) . Since B γ acts pointwisely, it’s easy to see that B γ ( P ˜ XP − P XP ) B − γ g = e γ | ( x − a, x − b, | Z R K ( x , y ) e − γ | ( y − a, y − b, | g ( y ) d y. Therefore, repeating similar steps gives us that:(18) ( A (1) γ g )( x ) = Z R K ( x , y ) e γ | ( x − a, x − b, | e − γ | ( y − a, y − b, | ˆ f (cid:18) x − y ∆ (cid:19) g ( y ) d y and so(( A (1) γ − A (1) ) g )( x ) = Z R K ( x , y )( e γ | ( x − a, x − b, | e − γ | ( y − a, y − b, | −
1) ˆ f (cid:18) x − y ∆ (cid:19) g ( y ) d y. Since we are interested in the spectral norm of A (1) γ − A (1) , we can use our expression for ( A (1) γ − A (1) ) g , take the inner product with any h ∈ L ( R ), and apply triangle inequality so that thebound to show is the following(19) Z R Z R (cid:12)(cid:12)(cid:12)(cid:12) h ( x ) K ( x , y )( e γ | ( x − a, x − b, | e − γ | ( y − a, y − b, | −
1) ˆ f (cid:18) x − y ∆ (cid:19) g ( y ) (cid:12)(cid:12)(cid:12)(cid:12) d y d x ≤ Cγ k g kk h k Using reverse triangle inequality and elementary calculus we have that: | e γ | ( x − a, x − b, | e − γ | ( y − a, y − b, | − | ≤ | e γ | ( x − y , x − y , − | − | = | e γ | x − y | − |≤ γ | x − y | e γ | x − y | . So since ˆ f is compactly supported on [ − ∆ , ∆] we conclude that we can bound Equation (19) withthe following:(20) γ ( √ e γ √ ) Z R Z R | h ( x ) K ( x , y ) g ( y ) | d y d x . For our final step, we can substitute in the definition of the kernel K from Equation (16) to concludethat: Z R Z R | h ( x ) K ( x , y ) g ( y ) | d y d x ≤ X m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| (cid:18) Z R | h ( x ) ψ ( j ) m,n ( x ) | d x (cid:19)(cid:18) Z R | g ( y ) ψ ( j ′ ) m ′ ,n ′ ( y ) | d y (cid:19) ≤ C k h kk g k , where the last inequality follows from Proposition 6.2. This proves that for all γ ≥ k A (1) γ − A (1) k ≤ (cid:16) C √ e γ √ (cid:17) γ which is what we wanted to show.7.1.2. Bounding k A (2) γ − A (2) k . Let’s begin by recalling the definitions for A (2) γ and A (2) : A (2) γ = Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ ( Q γ XP γ + P γ XQ γ ) e − i ( Xt + Y t ) / ∆ d t d t A (2) = Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ ( QXP + P XQ ) e − i ( Xt + Y t ) / ∆ d t d t Hence we can write the difference we’re interested in as: Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ ( Q γ XP γ − QXP + P γ XQ γ − P XQ ) e − i ( Xt + Y t ) / ∆ d t d t Due to our decay estimates on the projector P , by Corollary 5.1(iv), we have that for all γ sufficientlysmall k Q γ XP γ − QXP k = O ( γ ) , and k P γ XQ γ − P XQ k = O ( γ ) . Hence applying these estimates we have that we can find a constant
C > k A (2) γ − A (2) k ≤ Cγ (cid:18)Z | f ( t ) | d t (cid:19) (cid:18)Z | f ( t ) | d t (cid:19) finishing the proof that k A (2) γ − A (2) k = O ( γ ).7.2. Proof of Equation (13) . To begin this section, let us fix some µ ∈ R and recall the quantitywe want to bound:(13, revisited) ( X − µ + i ) − [ b X γ , Y ]( X − µ + i ) − . Similar to the previous section, our first step bounding this quantity will be to rewrite [ b X γ , Y ] intoa more friendly form. Using that [ X, Y ] = 0 we have that[ b X γ , Y ] = [ b X γ − X, Y ]= [ B γ ( b X − X ) B − γ , Y ] LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 17 = B γ [ b X − X, Y ] B − γ = Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ B γ [ ˜ X − X, Y ] B − γ e − i ( Xt + Y t ) / ∆ d t d t We can now decompose the difference ˜ X − X into two parts:˜ X − X = (cid:16) P ˜ XP − P XP (cid:17) + (cid:16) P XQ + QXP (cid:17) . Using this decomposition, let’s define ˜ A (1) and ˜ A (2) as follows:˜ A (1) := Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ B γ [ P ˜ XP − P XP, Y ] B − γ e − i ( Xt + Y t ) / ∆ d t d t ˜ A (2) := Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ B γ [ P XQ + QXP, Y ] B − γ e − i ( Xt + Y t ) / ∆ d t d t = Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ [ P γ XQ γ + Q γ XP γ , Y ] e − i ( Xt + Y t ) / ∆ d t d t With this notation, clearly( X − µ + i ) − [ b X γ , Y ]( X − µ + i ) − = ( X − µ + i ) − ( ˜ A (1) − ˜ A (2) )( X − µ + i ) − . Similar to Section 7.1, we will show that ( X − µ + i ) − ˜ A (1) ( X − µ + i ) − is bounded in Section 7.2.1and ( X − µ + i ) − ˜ A (2) ( X − µ + i ) − is bounded in Section 7.2.2.7.2.1. Bounding ˜ A (1) . We start by observing k ( X − µ + i ) − A (1) ( X − µ + i ) − k≤ k ( X − µ + i ) − kk A (1) kk ( X − µ + i ) − k≤ k A (1) k so it suffices to show that k A (1) k is bounded. Repeating similar calculations as given in Section 7.1.1one can easily calculate the action of ˜ A (1) for any g ∈ C ∞ c ( R ) as follows (cf. Equation (18)):( ˜ A (1) g )( x ) := Z R K ( x , y )( x − y ) e γ | ( x − a, x − b, | e − γ | ( y − a, y − b, | ˆ f (cid:18) x − y ∆ (cid:19) g ( y ) d y Next, taking the inner product with any h ∈ L ( R ) we have that: h h, ˜ A (1) g i≤ Z R Z R | h ( x ) || K ( x , y ) || x − y | e γ | ( x − a, x − b, | e − γ | ( y − a, y − b, | ˆ f (cid:18) x − y ∆ (cid:19) d y d x Since by reverse triangle inequality e γ | ( x − a, x − b, | e − γ | ( y − a, y − b, | ≤ e γ | ( x − y , x − y , − | = e γ | x − y | using the fact that ˆ f is supported on [ − ∆ , ∆] we conclude that h h, ˜ A (1) g i ≤ ∆ e γ √ Z R Z R | h ( x ) || K ( x , y ) || g ( y ) | d y d x Hence, applying Proposition 6.2 we conclude that k ˜ A (1) k = sup k h k = k g k =1 |h h, ˜ A (1) g i| ≤ C ∆ e γ √ completing the proof that ˜ A (1) is bounded. Bounding ˜ A (2) . Unlike the proof in the previous section, to show that ( X − µ + i ) − ˜ A (2) ( X − µ + i ) − is bounded, we will need to use the additional decay provided by ( X − µ + i ) − . In thissection we will prove the following bounds: k ( X − µ + i ) − [ P γ XQ γ , Y ]( X − µ + i ) − k = O (1)(21) k ( X − µ + i ) − [ Q γ XP γ , Y ]( X − µ + i ) − k = O (1) . (22)Proving these estimates show that ˜ A (2) is bounded by the following argument k ( X − µ + i ) − ˜ A (2) ( X − µ + i ) − k = k Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ ( X − µ + i ) − [ P γ XQ γ + Q γ XP γ , Y ] · ( X − µ + i ) − e − i ( Xt + Y t ) / ∆ d t d t k≤ k ( X − µ + i ) − [ P γ XQ γ + Q γ XP γ , Y ]( X − µ + i ) − k (cid:18)Z R | f ( t ) | d t (cid:19) < ∞ which proves Equation (13).We will only prove Equation (21) in this section, Equation (22) follows by similar steps. Usingthat P γ Q γ = Q γ P γ = 0 we have that for any µ ∈ R :[ P γ XQ γ , Y ] = [ P γ ( X − µ ) Q γ , Y ]= P γ ( X − µ ) Q γ Y − Y P γ ( X − µ ) Q γ = P γ ( X − µ )[ Q γ , Y ] + P γ ( X − µ ) Y Q γ − [ Y, P γ ]( X − µ ) Q γ − P γ Y ( X − µ ) Q γ where in the last line we have commuted Q γ and Y in the first term and P γ and Y in the secondterm. Since [ Y, X − µ ] = 0 we see that two of the terms above cancel so we get that:[ P γ XQ γ , Y ] = P γ ( X − µ )[ Q γ , Y ] − [ Y, P γ ]( X − µ ) Q γ Hence we have that k ( X − µ + i ) − [ P γ XQ γ , Y ]( X − µ + i ) − k = k ( X − µ + i ) − (cid:0) P γ ( X − µ )[ Q γ , Y ] − [ Y, P γ ]( X − µ ) Q γ (cid:1) ( X − µ + i ) − k≤ k ( X − µ + i ) − P γ ( X − µ ) kk [ Q γ , Y ] k + k [ Y, P γ ] kk ( X − µ ) Q γ ( X − µ + i ) − k . Due to Corollary 5.1(iii) we know that k [ P γ , Y ] k = k [ Q γ , Y ] k < ∞ hence to finish the bound, weonly need to show that k ( X − µ + i ) − P γ ( X − µ ) k and k ( X − µ ) Q γ ( X − µ + i ) − k are both bounded.This is easy to see however since( X − µ + i ) − P γ ( X − µ ) = ( X − µ + i ) − [ P γ , X ] + ( X − µ + i ) − ( X − µ ) P γ ( X − µ ) Q γ ( X − µ + i ) − = [ X, Q γ ]( X − µ + i ) − + Q γ ( X − µ )( X − µ + i ) − which are both clearly bounded due to Corollary 5.1(iii). This proves Equation (13) completingthe proof of Proposition 7.1. 8. P b XP has uniform spectral gaps Let us begin this section by first noting that since
P XP is essentially self-adjoint and k X − b X k isbounded (see Section 6), then that implies P b XP is essentially self-adjoint so the notion of uniformspectral gaps makes sense. In particular, we have the following easy lemma Lemma 8.1.
Suppose that P is a projector satisfying decay estimates as Assumption 4. Supposefurther that P admits a basis with finitely degenerate centers which is s -localized for some s > .Then P b XP is essentially self-adjoint. LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 19
Proof.
Recall the definitions of ˜ X and b X :˜ X = X m,n,j m | ψ ( j ) m,n ih ψ ( j ) m,n | + QXQ (7, revisited) b X = Z R f ( t ) f ( t ) e i ( Xt + Y t ) / ∆ ˜ Xe − i ( Xt + Y t ) / ∆ d t d t . (8, revisited)Since f is real valued and ˜ X is clearly a symmetric operator, it’s easy to see that b X is also asymmetric operator.Next, notice that P b XP = P XP + P ( b X − X ) P We have shown in previous work that
P XP is essentially self-adjoint under Assumption 4 (see [20,Section 5.1]). Since k P ( b X − X ) P k ≤ k b X − X k is bounded due to our proof from Section 6, by theKato-Rellich theorem [19, Theorem X.12], P b XP is essentially self-adjoint. (cid:3) Having established essential self-adjointness, the main goal of this section is to prove the followingproposition:
Proposition 8.2.
Suppose that P is a projector satisfying decay estimates as Assumption 4. Sup-pose further that P admits a basis with finitely degenerate centers which is s -localized for some s > / . Next, define a set of gaps G as follows: (23) G = [ m ∈ Z (cid:18) m + 14 , m + 34 (cid:19) . If b X is as defined in Equation (8) then for ∆ > sufficiently large, G ⊆ ρ ( P b XP ) . Hence for sucha choice of ∆ , P b XP has uniform spectral gaps. The basic idea behind proving Proposition 8.2 is to pick some λ ∈ G and consider ( λ − P b XP ) − .Since by construction σ ( P ˜ XP ) ⊆ Z we can formally write:( λ − P b XP ) − = ( λ − P ˜ XP + P ˜ XP − P b XP ) − = ( λ − P ˜ XP ) − (cid:16) I − ( P b XP − P ˜ XP )( λ − P ˜ XP ) − (cid:17) − . If we can show that for some constant C (24) k ( P b X P − P ˜ XP )( λ − P ˜ XP ) − k ≤ C ∆ − , then by picking ∆ ≥ (2 C ) − we have that k ( λ − P b XP ) − k ≤ k ( λ − P ˜ XP ) − kk (cid:16) I − ( P b XP − P ˜ XP )( λ − P ˜ XP ) − (cid:17) − k≤ (cid:16) (cid:17) − (cid:16) − (cid:17) − = 8 , where we have used that λ ∈ G and σ ( P ˜ XP ) ⊆ Z . Hence λ ∈ ρ ( P b XP ).While it is possible to prove the bound in Equation (24), we found proving this seems to require { ψ ( j ) m,n } is s -localized with s >
3. We can slightly improve this to s > / λ − P ˜ XP ) − “symmetrically”.For this let us define the square root of ( λ − P ˜ XP ) − . Explicitly, for any λ ∈ G we define S λ asfollows(25) S λ := | λ | − / Q + X m,n,j | λ − m | − / | ψ ( j ) m,n ih ψ ( j ) m,n | Note that by construction [ S λ , P ] = 0. Since P + Q = I and the collection { ψ ( j ) m,n } spans range ( P ) we have that: λ − P ˜ XP = λP + λQ − X m,n,j m | ψ ( j ) m,n ih ψ ( j ) m,n | = λQ + X m,n,j ( λ − m ) | ψ ( j ) m,n ih ψ ( j ) m,n | . A simple calculation shows that S λ ( λ − P ˜ XP ) S λ = λ | λ | Q + X m,n,j λ − m | λ − m | | ψ ( j ) m,n ih ψ ( j ) m,n | . Hence, since λ ∈ R , S λ ( λ − P ˜ XP ) S λ has eigenvalues ± S λ we can now repeat similar steps to before to get( λ − P b XP ) − = ( λ − P ˜ XP + P ˜ XP − P b XP ) − = S λ (cid:16) S λ ( λ − P ˜ XP ) S λ − S λ ( P b XP − P ˜ XP ) S λ (cid:17) − S λ . Therefore if we can show that(26) k S λ ( P b XP − P ˜ XP ) S λ k ≤ C ∆ − then by choosing ∆ ≥ (2 C ) the previous argument implies that λ ∈ ρ ( P b XP ).Let’s start our proof of Equation (26) by considering the difference P b XP − P ˜ XP . Using thefact that R f = 1 we have that P b XP − P ˜ XP = Z R f ( t ) f ( t ) P e i ( Xt + Y t ) / ∆ ˜ Xe − i ( Xt + Y t ) / ∆ P d t d t − P ˜ XP = Z R f ( t ) f ( t ) P (cid:16) e i ( Xt + Y t ) / ∆ ˜ Xe − i ( Xt + Y t ) / ∆ − ˜ X (cid:17) P d t d t . For the next few steps, let’s define the difference in parenthesis as D : D ( t , t ) := e i ( Xt + Y t ) / ∆ ˜ Xe − i ( Xt + Y t ) / ∆ − ˜ X. With this short-hand notation, we have that:(27) k S λ ( P b XP − P ˜ XP ) S λ k ≤ Z R | f ( t ) || f ( t ) |k S λ P D ( t , t ) P S λ k d t d t We will now use techniques similar to those used by Hastings [5] to control Equation (27). Oneimportant difference between the present work and previous work is that the operators
X, Y, ˜ X arenot bounded. Despite this fact, due to the multiplication on the left and right by S λ , we are ableto control Equation (27) and prove a similar bound to the one proved in Hastings’ work [5, Lemma1].Our first step of controlling Equation (27) will be exchange the decay provided by S λ (which isdiagonal in the basis { ψ ( j ) m,n } ) for h X − λ i − / (which is diagonal in the position basis). Formally,we calculate k S λ P D ( t , t ) P S λ k = k S λ P h X − λ i / h X − λ i − / D ( t , t ) h X − λ i − / h X − λ i / P S λ k≤ k S λ P h X − λ i / kkh X − λ i − / D ( t , t ) h X − λ i − / kkh X − λ i / P S λ k LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 21
Intuitively speaking, we should expect that k S λ P h X − λ i / k and kh X − λ i / P S λ k are both boundedsince S λ is the square root of ˜ X when restricted to range ( P ) and X and ˜ X differ by O (1) in thespectral norm. Indeed, we have the following lemma: Lemma 8.3.
Suppose that P is a projector satisfying decay estimates as Assumption 4. Supposefurther that P admits a basis with finite degenerate centers which is s -localized for some s > .Then there exists a constant C > such that for any λ ∈ G : k S λ P h X − λ i / k ≤ C kh X − λ i / P S λ k ≤ C Proof.
Given in Appendix E. (cid:3)
Combining this lemma with the above calculation and Equation (27) we therefore conclude that k S λ ( P b XP − P ˜ XP ) S λ k≤ C Z R | f ( t ) || f ( t ) |kh X − λ i − / D ( t , t ) h X − λ i − / k d t d t (28)For the next few steps, let us define the shorthand(29) ˜ X b := h X − λ i − / ˜ X h X − λ i − / . Since k ˜ X − X k = O (1) it’s easy to see that for a fixed value of λ , the operator ˜ X b is bounded asan operator acting from L ( R ) → L ( R ). The subscript b is intended to be suggestive of the factthat ˜ X b is a bounded version of ˜ X .We can write the quantity h X − λ i − / D ( t , t ) h X − λ i − / in terms of ˜ X b by commuting h X − λ i − / with e i ( Xt + Y t ) / ∆ and e − i ( Xt + Y t ) / ∆ as follows: h X − λ i − / D ( t , t ) h X − λ i − / = h X − λ i − / (cid:18) e i ( Xt + Y t ) / ∆ ˜ Xe − i ( Xt + Y t ) / ∆ − ˜ X (cid:19) h X − λ i − / = e i ( Xt + Y t ) / ∆ ˜ X b e − i ( Xt + Y t ) / ∆ − ˜ X b . Therefore, defining A ( t , t ) as A ( t , t ) = e i ( Xt + Y t ) / ∆ ˜ X b e − i ( Xt + Y t ) / ∆ we see that by definition h X − λ i − / D ( t , t ) h X − λ i − / = A ( t , t ) − A (0 , . We now state an important proposition regarding ˜ X b : Proposition 8.4.
Suppose that P is a projector satisfying decay estimates as Assumption 4. Sup-pose further that P admits a basis with finitely degenerate centers which is s -localized for some s > / . Then for any λ ∈ G there exists a finite constant C > such that k [ X, ˜ X b ] k = kh X − λ i − / [ X, ˜ X ] h X − λ i − / k ≤ C, k [ Y, ˜ X b ] k = kh X − λ i − / [ Y, ˜ X ] h X − λ i − / k ≤ C. (30) Proof.
Given in Appendix F. (cid:3)
With this proposition in mind, for any φ ∈ C ∞ c ( R ) we differentiate A ( t , t ) φ with respect to t to get: ∂ t A ( t , t ) φ = i ∆ − e i ( Xt + Y t ) / ∆ (cid:0) X ˜ X b − ˜ X b X (cid:1) e − i ( Xt + Y t ) / ∆ φ = i ∆ − e i ( Xt + Y t ) / ∆ [ X, ˜ X b ] e − i ( Xt + Y t ) / ∆ φ. This differentiation step is justified for any φ ∈ C ∞ c ( R ) since ˜ X b X and X ˜ X b are both boundedoperators on C ∞ c ( R ). The fact that ˜ X b X is bounded is clear since X is bounded on C ∞ c ( R ) and˜ X b , as defined in (29), is a bounded operator. The fact that X ˜ X b is bounded follows from theidentity X ˜ X b = [ X, ˜ X b ] + ˜ X b X which is bounded due to Proposition 8.4.An analogous argument shows that ∂ t A ( t , t ) φ = i ∆ − e i ( Xt + Y t ) / ∆ [ Y, ˜ X b ] e − i ( Xt + Y t ) / ∆ φ. Due to Proposition 8.4, it’s easy to check that both ∂ t A ( t , t ) φ and ∂ t A ( t , t ) φ are continuousfunctions of t , t so we can apply mean value theorem to conclude there exists a ( c , c ) ∈ [0 , t ] × [0 , t ] so that: k ( A ( t , t ) − A (0)) φ k ≤ ∆ − | c |k e i ( Xc + Y c ) / ∆ [ X, ˜ X b ] e − i ( Xc + Y c ) / ∆ φ k + ∆ − | c |k e i ( Xc + Y c ) / ∆ [ Y, ˜ X b ] e − i ( Xc + Y c ) / ∆ φ k≤ ∆ − (cid:16) | t |k [ X, ˜ X b ] k + | t |k [ Y, ˜ X b ] k (cid:17) k φ k Since C ∞ c ( R ) is dense in L ( R ), this implies that there exists a finite constant C so that kh X − λ i − / D ( t , t ) h X − λ i − / k ≤ C ∆ − ( | t | + | t | ) . Hence, substituting this bound into Equation (28), we have that k S λ ( P b XP − P ˜ XP ) S λ k≤ C ∆ − Z R | f ( t ) || f ( t ) | ( | t | + | t | ) d t d t ≤ C ′ ∆ − , where to get the last line we have used the fact that by construction f ( t ) , tf ( t ) ∈ L ( R ). Thiscompletes the proof of Proposition 8.2 and hence establishes that for ∆ sufficiently large P b XP hasuniform spectral gaps.9. The Band Projectors for P b XP Satisfy the Assumptions of Theorem 2
Over the past few sections, we have proved a number of important properties of the operator b X .In particular we have shown that:(1) b X is close to X (Section 6) k b X − X k = O (1)(2) b X is exponentially localized (Section 7)(a) k b X γ − b X k = O ( γ )(b) ∀ µ ∈ R , k ( X − µ + i ) − [ b X γ , Y ]( X − µ + i ) − k = O (1)(3) P b XP has uniform spectral gaps (Section 8) [ m ∈ Z (cid:18) m + 14 , m + 34 (cid:19) ⊆ ρ ( P b XP )Our next step will be to use all of these properties to show that if { P j } j ∈ Z are the band projectorsfor P b XP , then these projectors satisfy the assumptions of Theorem 2. LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 23
First, due to the fact that P b XP has uniform spectral gaps, we can define the orthogonal projec-tion P j by the Riesz formula(31) P j := 12 πi P Z C j ( λ − P b XP ) − d λ ! P where the contour C j is chosen so that C j encloses the integer j and avoids σ ( P b XP ). We willassume without loss of generality that the contour C j is chosen so that for all j ∈ Z if λ ∈ C j then | λ − j | ≤
1. We can assume this without loss of generality due to the characterization of ρ ( P b XP )proven in Section 8.As a technical note, the addition of P in the definition of P j makes it so that range ( P j ) ⊆ range ( P ) for all j ∈ Z and ensures that P j P j = P . Since P is a projection, we know that[ P b XP, P ] = 0 and hence P j can be equivalently defined as(32) P j := 12 πi Z C j ( λ − P b XP ) − P d λ. The most technically involved part of showing that P j satisfies the estimates of Theorem 2 isshowing that the resolvent ( λ − P b XP ) − is exponentially localized in following sense: Proposition 9.1.
Suppose that P satisfies decay estimates as Assumption 4. There exists finite,positive constants ( C , γ ∗ ) such that for all ≤ γ ≤ γ ∗ , all j ∈ Z , and all λ ∈ C j we have k B γ ( λ − P b XP ) − B − γ k = k ( λ − P γ b X γ P γ ) − k ≤ C Once we prove Proposition 9.1, we can then prove that P j satisfies the assumptions of Theorem 2.More specifically, we can show that there exists finite, positive constants ( C , C , C ) such that forall j , P j satisfies the following estimates for all γ sufficiently small: k P j,γ − P j k ≤ C γ (33a) k [ P j,γ , Y ] k ≤ C (33b) k P j,γ ( X − j ) k ≤ C (33c) k ( X − j ) P j,γ k ≤ C (33d)It is important to note that the constants ( C , C , C , C ) from Proposition 9.1 and Equation(33) are independent of the choice j . While the tools from our previous work [20] allow one toprove these bounds uniformly in j , there are a few subtle technical issues which arise as part ofthese proofs. For clarity of presentation, in the next few sections we will prove Proposition 9.1 andEquation (33) for j = 0 only. We will then return to discuss the modifications needed to prove thebounds uniformly for j = 0 in Section 9.6.The remainder of this section is structured as follows. We begin by proving an important technicallemma in Section 9.1. We then make use of this lemma in Section 9.2 to prove Proposition 9.1 for j = 0. After proving this proposition, we will prove Equation (33a) in Section 9.3, Equation (33b)in Section 9.4, and Equations (33c) and (33d) in Section 9.5 (all for j = 0). Finally, in Section 9.6,we will show how to generalize the previous proofs to j = 0. Once we have shown all of theseestimates, Theorem 2 then implies Theorem 1 completing the proof of our main theorem.9.1. Projected Position Operators and Their Resolvents.
Over the next few sections, wewill perform a simple calculation which we will repeat multiple times. Instead of repeating thiscalculation each time, in this section, we will show this calculation in detail with an illustrativeexample. We will then state a technical lemma (Lemma 9.2) which extracts the essence of thecalculation at the end of this section.
For our illustrative example we will show how to bound the following for λ ∈ C j when j = 0:(34) XP ( λ − P b XP ) − . Intuitively, XP ( λ − P b XP ) − should be bounded since b X is close to X and ( λ − P b XP ) − “inverts” b X on range ( P ).Formally, we calculate k XP ( λ − P b XP ) − k = k ( X − b X + b X ) P ( λ − P b XP ) − k≤ k X − b X kk ( λ − P b XP ) − k + k b XP ( λ − P b XP ) − k . Since we know that k X − b X k = O (1), to show that Equation (34) is bounded we only need to showthat b XP ( λ − P b XP ) − is bounded.Using the fact that I = Q + P we have that k b XP ( λ − P b XP ) − k = k ( Q + P ) b XP ( λ − P b XP ) − k≤ k Q b XP kk ( λ − P b XP ) − k + k P b XP ( λ − P b XP ) − k The first of these two terms is bounded since k ( λ − P b XP ) − k is bounded and k Q b XP k ≤ k Q ( b X − X ) P k + k QXP k≤ k b X − X k + k QXP k where k QXP k is bounded by Corollary 5.1(ii) with γ = 0. Finally P b XP ( λ − P b XP ) − is boundedsince k P b XP ( λ − P b XP ) − k = k ( P b XP − λ + λ )( λ − P b XP ) − k = k − I + λ ( λ − P b XP ) − k≤ | λ |k ( λ − P b XP ) − k . (35)The final bound in Equation (35) illustrates the importance of our choice of λ ∈ C j where j = 0.Since λ ∈ C , by construction we know that | λ | = | λ − | ≤
1. Therefore, k P b XP ( λ − P b XP ) − k ≤ k ( λ − P b XP ) − k which is a constant independent of the choice of j . If we were considering j = 0, then typicallywe would have that | λ | ≫ j = 0.From this calculation, we conclude that XP ( λ − P b XP ) − is bounded by a constant only depend-ing on k QXP k , k X − b X k , and the gap size of P b XP . By repeating this calculation multiple timesusing our decay estimates on P (Assumption 4 and Corollary 5.1), we have the following lemma: Lemma 9.2.
Suppose that P is a projector satisfying decay estimates as Assumption 4. Thereexists finite, positive constants ( C , C , C , C ) such that for all λ ∈ C j where j = 0 :(i) k ( λ − P b XP ) − P b X k ≤ C (ii) k b XP ( λ − P b XP ) − k ≤ C (iii) k ( λ − P γ b X γ P γ ) − P γ b X γ k ≤ C (iv) k ( λ − P γ b X γ P γ ) − P γ X k ≤ C Proof of Proposition 9.1.
The main goal of this section is to show that there exists a finite,positive constant C such that for j = 0 we have:sup λ ∈C j k ( λ − P γ b X γ P γ ) − k ≤ C. This bound is proved by the following chain of implications which hold for all λ ∈ C j (where USGis an abbreviation for uniform spectral gaps): P b XP has USG = ⇒ k ( λ − P b XP ) − k < ∞ = ⇒ k ( λ − P b XP γ ) − k < ∞ = ⇒ k ( λ − P γ b XP γ ) − k < ∞ = ⇒ k ( λ − P γ b X γ P γ ) − k < ∞ . (36)In words, since P b XP has uniform spectral gaps, we know that k ( λ − P b XP ) − k < ∞ for all λ ∈ C j . Using this fact, along with our previous estimates on P and b X , we can conclude that k ( λ − P b XP γ ) − k < ∞ for all γ sufficiently small. Once we know that ( λ − P b XP γ ) − is bounded,we can use that estimate to show that ( λ − P γ b XP γ ) − is bounded for all γ sufficiently small.Finally, once we know that ( λ − P γ b XP γ ) − is bounded, we can show that ( λ − P γ b X γ P γ ) − isbounded, finishing the proof of the proposition. These steps will be detailed below.9.2.1. Bounding k ( λ − P b XP γ ) − k . We start by performing a formal calculation. Adding and sub-tracting P b XP in the resolvent we want to bound gives that:( λ − P b XP γ ) − = ( λ − P b XP + P b XP − P b XP γ ) − = (cid:16) ( λ − P b XP ) − P b X ( P γ − P ) (cid:17) − = (cid:16) I − ( λ − P b XP ) − P b X ( P γ − P ) (cid:17) − ( λ − P b XP ) − . Now observe that k ( λ − P b XP ) − P b X ( P γ − P ) k≤ k ( λ − P b XP ) − P b X kk P γ − P k . Due to Lemma 9.2(i), we know that ( λ − P b XP ) − P b X is bounded. We also know that that k P γ − P k = O ( γ ) by Assumption 4(i) and thus we can pick γ sufficiently small so that k ( λ − P b XP ) − P b X ( P γ − P ) k ≤ . Therefore, by Neumann series we conclude that for all γ sufficiently small k ( λ − P b XP γ ) − k ≤ k (cid:16) I − ( λ − P b XP ) − P b X ( P γ − P ) (cid:17) − ( λ − P b XP ) − k≤ (cid:16) − (cid:17) − k ( λ − P b XP ) − k which is bounded by a constant due to the fact that P b XP has uniform spectral gaps and the choiceof λ .9.2.2. Bounding k ( λ − P γ b XP γ ) − k . Now that we’ve shown that ( λ − P b XP γ ) − is bounded, we canbound ( λ − P γ b XP γ ) − . Performing similar formal calculations to before( λ − P γ b XP γ ) − = ( λ − P b XP γ + P b XP γ − P γ b XP γ ) − = (cid:16) ( λ − P b XP γ ) − ( P γ − P ) b XP γ (cid:17) − = ( λ − P b XP γ ) − (cid:16) I − ( P γ − P ) b XP γ ( λ − P b XP γ ) − (cid:17) − . Similar to before, we will want to show that b XP γ ( λ − P b XP γ ) − is bounded by a constant anduse that k P γ − P k = O ( γ ) to conclude that ( λ − P γ b XP γ ) − is bounded for all γ sufficientlysmall. Unfortunately, the calculation used to prove Lemma 9.2 does not apply here. It turns outhowever that a slight modification of the calculation which proves Lemma 9.2 allows us to showthat b XP γ ( λ − P XP γ ) − is bounded.First, let us define the shorthand E := P γ − P = Q − Q γ . Observe that Q − Q γ = E ⇒ Q = E + Q γ . Using the fact that I = P + Q we have that b XP γ ( λ − P b XP γ ) − = ( P + Q ) b XP γ ( λ − P b XP γ ) − = ( P + Q γ + E ) b XP γ ( λ − P b XP γ ) − = ( P + Q γ ) b XP γ ( λ − P b XP γ ) − + E b XP γ ( λ − P b XP γ ) − Moving the term multiplied by E to the left hand side then gives that:( I − E ) b XP γ ( λ − P b XP γ ) − = ( P + Q γ ) b XP γ ( λ − P b XP γ ) − b XP γ ( λ − P b XP γ ) − = ( I − E ) − ( P + Q γ ) b XP γ ( λ − P b XP γ ) − . Note that since k E k = k P γ − P k = O ( γ ), we can pick γ sufficiently small so that k ( I − E ) − k isbounded. Therefore, taking norms on both sides gives that k b XP γ ( λ − P b XP γ ) − k≤ k ( I − E ) − k (cid:16) k P b XP γ ( λ − P b XP γ ) − k + k Q γ b XP γ kk ( λ − P b XP γ ) − k (cid:17) This final equation can be easily be seen to be bounded by constant by observing that k P b XP γ ( λ − P b XP γ ) − k ≤ | λ |k ( λ − P b XP γ ) − kk Q γ b XP γ k ≤ k Q γ ( X − b X ) P γ k + k Q γ XP γ k and Corollary 5.1(i,ii). Notice again, we require that λ ∈ C j for j = 0 to conclude that | λ | ≤ j .9.2.3. Bounding k ( λ − P γ b X γ P γ ) − k . Now that we’ve shown ( λ − P γ b XP γ ) − is bounded, we canfinally finish the proof of Proposition 9.1. Similar to before, we have that( λ − P γ b X γ P γ ) − = ( λ − P γ b XP γ + P γ b XP γ − P γ b X γ P γ ) − = (cid:16) ( λ − P γ XP γ ) − P γ ( b X γ − b X ) P γ (cid:17) − = (cid:16) I − ( λ − P γ XP γ ) − P γ ( b X γ − b X ) P γ (cid:17) − ( λ − P γ XP γ ) − . Since k P γ ( b X γ − b X ) P γ k ≤ k P γ k k b X γ − b X k and in Section 7 we showed that k b X γ − b X k = O ( γ ), we can choose γ sufficiently small so that k ( λ − P γ XP γ ) − P γ ( b X γ − b X ) P γ k ≤ γ sufficiently small( λ − P γ b X γ P γ ) − ≤ (cid:16) − (cid:17) − k ( λ − P γ XP γ ) − k . LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 27
Combined with the chain of implications in Equation (36), we conclude that (for j = 0)sup λ ∈C j k ( λ − P γ b X γ P γ ) − P γ k < ∞ which finishes the proof of Proposition 9.1.9.3. Proof of Equation (33a) . Similar to the previous section, let us define the shorthand E := P γ − P . We begin by writing down the definition of P j,γ and P j as contour integrals P j,γ − P j = 12 πi Z C j ( λ − P γ b X γ P γ ) − P γ − ( λ − P b XP ) − P d λ = 12 πi Z C j (cid:16) ( λ − P γ b X γ P γ ) − − ( λ − P b XP ) − (cid:17) P γ + ( λ − P b XP ) − E d λ Taking norms of both sides then gives that: k P j,γ − P j k≤ ℓ ( C j )2 π sup λ ∈C j (cid:16) k ( λ − P γ b X γ P γ ) − − ( λ − P b XP ) − kk P γ k + k ( λ − P b XP ) − kk E k (cid:17) . Since λ ∈ C j and k E k = O ( γ ), we know that k ( λ − P b XP ) − kk E k = O ( γ ). Therefore, to finish theproof of Equation (33a) we only need to show the first term is O ( γ ). Applying the second resolventidentity gives ( λ − P γ b X γ P γ ) − − ( λ − P b XP ) − = ( λ − P γ b X γ P γ ) − ( P γ b X γ P γ − P b XP )( λ − P b XP ) − . Next, adding and subtracting various terms gives( λ − P γ b X γ P γ ) − ( P γ b X γ P γ − P b XP )( λ − P b XP ) − = ( λ − P γ b X γ P γ ) − (cid:16) P γ b X γ P γ − (cid:0) P γ b X γ P + P γ b X γ P (cid:1) − P b XP (cid:17) ( λ − P b XP ) − = ( λ − P γ b X γ P γ ) − (cid:16) P γ b X γ E + P γ b X γ P + (cid:0) P γ b XP − P γ b XP (cid:1) − P b XP (cid:17) ( λ − P b XP ) − = ( λ − P γ b X γ P γ ) − (cid:0) P γ b X γ E + P γ ( b X γ − b X ) P + E b XP (cid:1) ( λ − P b XP ) − The middle term, P γ ( b X γ − b X ) P is clearly O ( γ ) since we know that k b X γ − b X k = O ( γ ). Since k E k = O ( γ ) to finish the lemma we only need to show that( λ − P γ b X γ P γ ) − P γ b X γ b XP ( λ − P b XP ) − are both bounded operators. But we already argued that these quantities are bounded in Lemma 9.2(ii,iii).Hence Equation (33a) is proved.9.4. Proof of Equation (33b) . We begin this proof by writing P j,γ in terms of its contour integraland taking operator norms: k [ P j,γ , Y ] k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" πi Z C j ( λ − P γ b X γ P γ ) − P γ d λ ! , Y ≤ ℓ ( C j )2 π sup λ ∈C j k [( λ − P γ b X γ P γ ) − P γ , Y ] k . Therefore, to prove Equation (33b) it’s enough to show that [( λ − P γ b X γ P γ ) − P γ , Y ] is bounded.Expanding this commutator and using the fact that [( λ − P γ b X γ P γ ) − , P γ ] = 0, we have that[( λ − P γ b X γ P γ ) − P γ , Y ]= ( λ − P γ b X γ P γ ) − P γ Y − Y ( λ − P γ b X γ P γ ) − P γ = ( λ − P γ b X γ P γ ) − P γ Y − Y P γ ( λ − P γ b X γ P γ ) − = ( λ − P γ b X γ P γ ) − P γ Y ( P γ + Q γ ) − ( P γ + Q γ ) Y P γ ( λ − P γ b X γ P γ ) − = [( λ − P γ b X γ P γ ) − , P γ Y P γ ]+ ( λ − P γ b X γ P γ ) − P γ Y Q γ − Q γ Y P γ ( λ − P γ b X γ P γ ) − Due to Corollary 5.1(ii) we know that Q γ Y P γ and P γ Y Q γ are both bounded. Since ( λ − P γ b X γ P γ ) − is also bounded, we therefore only need to control the first term. A standard commutator identityshows that: [( λ − P γ b X γ P γ ) − , P γ Y P γ ]= ( λ − P γ b X γ P γ ) − [ P γ b X γ P γ , P γ Y P γ ]( λ − P γ b X γ P γ ) − (37)Now we can rewrite the commutator [ P γ b X γ P γ , P γ Y P γ ] as follows:[ P γ b X γ P γ , P γ Y P γ ] = P γ b X γ P γ Y P γ − P γ Y P γ b X γ P γ = P γ b X γ ( I − Q γ ) Y P γ − P γ Y ( I − Q γ ) b X γ P γ = P γ [ b X γ , Y ] P γ − P γ b X γ Q γ Y P γ + P γ Y Q γ b X γ P γ Substituting this identity back into Equation (37) and taking norms then gives that(38) k ( λ − P γ b X γ P γ ) − [ P γ b X γ P γ , P γ Y P γ ]( λ − P γ b X γ P γ ) − k≤ k ( λ − P γ b X γ P γ ) − P γ [ b X γ , Y ] P γ ( λ − P γ b X γ P γ ) − k + k ( λ − P γ b X γ P γ ) − k k P γ b X γ Q γ kk Q γ Y P γ k + k ( λ − P γ b X γ P γ ) − k k P γ Y Q γ kk Q γ b X γ P γ k . The last two terms can easily be seen to be bounded using Corollary 5.1. In particular, k P γ b X γ Q γ k = k P γ ( b X γ − b X + b X − X + X ) Q γ k≤ k P γ kk b X γ − b X kk Q γ k + k P γ kk b X − X kk Q γ k + k P γ XQ γ k . Hence, since we’ve shown that k b X γ − b X k = O ( γ ), k b X − X k = O (1), and Corollary 5.1(i,ii), weconclude that the last two terms in (38) are bounded. Therefore, to finish the proof of this lemmawe only need to bound the first term in (38).To bound this term, we will appeal to one of the bounds proven in Section 7. In particular, inthat section we showed that (Equation (13)) for any µ ∈ R we have that(13, revisited) k ( X − µ + i ) − [ b X γ , Y ]( X − µ + i ) − k < ∞ Now let’s recall the quantity we want to bound( λ − P γ b X γ P γ ) − P γ [ b X γ , Y ] P γ ( λ − P γ b X γ P γ ) − . Our first step will be to exchange the decay provided by ( λ − P γ b X γ P γ ) − for ( X − µ + i ) − so thatwe can apply Equation (13). To perform this exchange, observe that for any µ ∈ R :( λ − P γ b X γ P γ ) − P γ = ( λ − P γ b X γ P γ ) − P γ ( X − µ + i )( X − µ + i ) − . LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 29
Now for any µ ∈ R we have k ( λ − P γ b X γ P γ ) − P γ ( X − µ + i ) k≤ k ( λ − P γ b X γ P γ ) − P γ ( X − µ ) k + k ( λ − P γ b X γ P γ ) − kk P γ k . Choosing µ = j = 0 and appealing to Lemma 9.2 we conclude there exists a constant C so that k ( λ − P γ b X γ P γ ) − P γ ( X + i ) k ≤ C. Similarly, k ( X + i ) P γ ( λ − P γ b X γ P γ ) − k ≤ C Hence, k ( λ − P γ b X γ P γ ) − P γ [ b X γ , Y ] P γ ( λ − P γ b X γ P γ ) − k = k ( λ − P γ b X γ P γ ) − P γ ( X + i )( X + i ) − [ b X γ , Y ] P γ ( λ − P γ b X γ P γ ) − k≤ C k ( X + i ) − [ b X γ , Y ] P γ ( λ − P γ b X γ P γ ) − k≤ C k ( X + i ) − [ b X γ , Y ]( X + i ) − k where to the get the last line we have inserted ( X + i ) − ( X + i ) on the right as we did on the left.This final line is bounded due to Equation (13), completing the proof of Equation (33b).9.5. Proof of Equations (33c) and (33d) . Since we are considering the special case where j = 0,the quantity to show are bounded are P j,γ X and XP j,γ . We we only show k P j,γ X k is bounded, theother bound follows via a similar argument. Writing P j,γ in terms of its contour integral gives k P j,γ X k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) πi Z C j ( λ − P γ b X γ P γ ) − P γ X d λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ℓ ( C j )2 π sup λ ∈C j k ( λ − P γ b X γ P γ ) − P γ X k But k ( λ − P γ b X γ P γ ) − P γ X k is bounded due to Lemma 9.2. This completes the proof of Equations(33c) and (33d).9.6. The Shifting Lemma.
In this section, we review some of the tools which were proven in [20]to prove Proposition 9.1 and Equation (33) with bounds independent of the choice j . The key trickis the following simple calculation. Suppose that λ ∈ C j for some j = 0. We have that( λ − P b XP ) − = (cid:16) ( λ − j + j ) − P ( b X − j + j ) P (cid:17) − = (cid:16) ( λ − j ) − P ( b X − j ) P + j ( I − P ) (cid:17) − = (cid:16) ( λ − j ) − P ( b X − j ) P + jQ (cid:17) − Now since
P Q = QP = 0 it can be shown that( λ − P b XP ) − P = (cid:16) ( λ − j ) − P ( b X − j ) P + jQ (cid:17) − P = (cid:16) ( λ − j ) − P ( b X − j ) P (cid:17) − P (39)Through this trick, we are able to effectively “shift down” the contour C j by j . More precisely,using this calculation, in [20, Appendix C] we proved the following lemma: Lemma 9.3 (Shifting Lemma) . Suppose P satisfies decay estimates as Assumption 4 and supposethat P XP has uniform spectral gaps with decomposition { σ j } j ∈ Z . For arbitrary η ∈ C , define λ η := λ − η and X η := X − η . Then the following are equivalent for all γ sufficiently small:(1) There exists a C > , independent of j , such that sup λ ∈C j k ( λ − P γ XP γ ) − k ≤ C (2) There exists a C ′ > , independent of j , such that sup λ ∈C j sup η ∈ σ j k ( λ η − P γ X η P γ ) − k ≤ C ′ Furthermore, if k ( λ − P γ XP γ ) − k is bounded we have for any j ∈ Z and η ∈ σ ( P XP ) : ( λ − P γ XP γ ) − P γ = ( λ η − P γ X η P γ ) − P γ . Since the proof of Lemma 9.3 in [20] only relies on the shifting calculation in Equation (39),Lemma 9.3 also holds if we replace X with b X . Choosing η = j we have that by the lemma for all γ sufficiently small P j,γ = Z C j ( λ − P γ b X γ P γ ) − P γ d λ = Z C j (cid:16) ( λ − j ) − P γ ( b X γ − j ) P γ (cid:17) − P γ d λ By replacing λ with λ − j and b X with b X − j in all of the arguments in Sections 9.2, 9.3, 9.4, 9.5 itcan be checked that instead of a dependence on | λ | we have a dependence | λ − j | which is boundedby the construction of C j . Therefore, we conclude that Proposition 9.1 and Equation (33) holdswith constants uniform in j proving the main theorem. Appendix A. Proof of Lemma 4.2
Let’s recall the statement we want to prove:
Lemma 4.2.
Suppose that P ∈ B ( L ( R )) is an orthogonal projector. Suppose further that { ψ α } and { φ α } are two distinct orthonormal bases for range ( P ) and both bases are s -localized with s > .If { ψ α } has finitely degenerate centers then { φ α } must also have finitely degenerate centers.Proof. For this proof let χ L, ( x ,y ) denote the characteristic function χ L, ( x ,y ) = x ∈ (cid:20) x − L , x + L (cid:21) × (cid:20) y − L , y + L (cid:21) . and let k · k F denote the Fr¨obenius norm (Schatten 2-norm). Since ψ α has finitely degeneratecenters we can relabel it as { ψ ( j ) m,n } as we did in Section 4.1 where ( m, n ) ∈ Z and j ∈ { , · · · , M } . Since { ψ ( j ) m,n } forms an orthonormal basis, for any ( x , y ) ∈ R by the definition of the Fr¨obeniusnorm we have that: k χ L, ( x ,y ) P k F = X m,n,j k χ L, ( x ,y ) P ψ ( j ) m,n k = M X j =1 X m,n k χ L, ( x ,y ) ψ ( j ) m,n k Our basic strategy of this proof will be to first show that if { ψ ( j ) m,n } is an s -localized basis with s > k χ L, ( x ,y ) P k F = O ( L ). This estimate essentially puts a limiton how strongly the center points for a localized bases from range ( P ) can cluster. If a basis for LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 31 range ( P ) does not satisfy the finitely degenerate centers condition, then its center points clusterarbitrarily strongly, which allows us to violate the O ( L ) bound leading to a contradiction.We’ll start by proving the estimate for k χ L, ( x ,y ) P k F . Towards these ends, let’s split the sumover m, n into two parts A := Z ∩ supp( χ L, ( x ,y ) ) A c := Z \ A. Note that the side length in the definition of A is 2 L not L . Since 2 L > L , we have the followingestimate for all ( m, n ) ∈ A c k χ L, ( x ,y ) ψ ( j ) m,n k = Z R χ L, ( x ,y ) ( x ) | ψ ( j ) m,n ( x ) | d x = Z R χ L, ( x ,y ) ( x ) h x − ( m, n ) i ǫ ) h x − ( m, n ) i ǫ ) | ψ ( j ) m,n ( x ) | d x ≤ h ( m − L, n − L ) i ǫ ) Z R χ L, ( x ,y ) ( x ) h x − ( m, n ) i ǫ ) | ψ ( j ) m,n ( x ) | d x ≤ C (1 + ( m − L ) + ( n − L ) ) ǫ . where in the second to last line we have used the following pointwise inequality which holds for all( m, n ) ∈ A c χ L, ( x ,y ) ( x ) h x − ( m, n ) i ǫ ) ≤ h ( m − L, n − L ) i ǫ ) and in the final line we have used our assumption that ψ m,n is s -localized for some s > ǫ sufficiently small so that Z R h x − ( m, n ) i ǫ ) | ψ ( j ) m,n ( x ) | d x ≤ C. Using this estimate we have that k χ L, ( x ,y ) P k F = M X j =1 X m,n k χ L, ( x ,y ) ψ ( j ) m,n k = M X j =1 X ( m,n ) ∈ A k χ L, ( x ,y ) ψ ( j ) m,n k + M X j =1 X ( m,n ) ∈ A c k χ L, ( x ,y ) ψ ( j ) m,n k ≤ M X j =1 X ( m,n ) ∈ A M X j =1 X ( m,n ) ∈ A c C (1 + ( m − L ) + ( n − L ) ) ǫ ) ≤ M L + X ( m,n ) ∈ Z CM (1 + ( m − L ) + ( n − L ) ) ǫ ) ≤ M L + C ′ where C ′ is an absolute constant independent of L .Next, towards a contradiction, let’s suppose that the basis { φ α } does not have finitely degeneratecenters. By definition, this means that for every N ∈ Z + we can find a ball B ( x N , y N ) so that this ball has more than N center points in it. Recall our notation for the unit box centered at ( m, n ): S m,n := (cid:20) m − , m + 12 (cid:19) × (cid:20) n − , n + 12 (cid:19) Using basic planar geometry, it’s easy to see that each ball of radius 1 overlaps with at most 4 ofthese unit boxes. Therefore, for each ( x N , y N ) (a center point for a box) we can also find a pair ofintegers ( m N , n N ) ∈ Z so that the box S m N ,n N contains at least N center points.While we are assuming that { φ α } does not have finitely degenerate centers, since this basis is s -localized with s >
1, we can still relabel this basis as { φ ( j ) m,n } where the degeneracy index j mayrun through an infinite number values.Since φ ( j ) m,n is s -localized for some s >
1, we also have that for all ( m, n, j ) k (1 − χ L, ( m,n ) ) φ ( j ) m,n k = Z R (1 − χ L, ( m,n ) ( x )) | φ ( j ) m,n ( x ) | d x = Z R (1 − χ L, ( m,n ) ( x )) h x − ( m, n ) i h x − ( m, n ) i | φ ( j ) m,n ( x ) | d x ≤ √ L Z R h x − ( m, n ) i | φ ( j ) m,n ( x ) | d x ≤ C ′′ √ L . where in the last line we have used the fact that φ ( j ) m,n is s -localized with s >
1. Since the constant C ′′ is uniform in ( m, n, j ), we may find a real number L ∗ so that for all ( m, n, j ) k (1 − χ L ∗ , ( m,n ) ) φ ( j ) m,n k ≤ ⇒ k χ L ∗ , ( m,n ) φ ( j ) m,n k ≥ . Since by assumption { φ ( j ) m,n } does not have finitely degenerate centers, there must be an index( m ∗ , n ∗ ) ∈ Z so that there are more than 4(4 M ( L ∗ ) + C ′ ) center points in the box S m ∗ ,n ∗ . Thisis a contradiction as k χ L ∗ , ( m ∗ ,n ∗ ) P k F = X m,n,j k χ L ∗ , ( m ∗ ,n ∗ ) φ ( j ) m,n k ≥ ⌈ M ( L ∗ ) + C ′ ) ⌉ X j =1 k χ L ∗ , ( m ∗ ,n ∗ ) φ ( j ) m ∗ ,n ∗ k ≥ M ( L ∗ ) + C ′ )which violates our previous bound. Hence { φ α } must also have finitely degenerate centers. (cid:3) Appendix B. P Estimates
In this section we will show that if H is a Hamiltonian satisfying Assumptions 1 and 2 and if P isa spectral projector onto σ then P satisfies the decay estimates from Assumption 4. That is thereexist finite, positive constants ( C , C , C , C , C , γ ∗ ) such that for all 0 ≤ γ ≤ γ ∗ the followingbounds hold:(i) k P γ − P k ≤ C γ (ii) (a) k [ P γ , X ] k ≤ C (b) k [ P γ , Y ] k ≤ C (iii) (a) k [ P γ − P, X ] k ≤ C γ (b) k [ P γ − P, Y ] k ≤ C γ (iv) k [ P, h X − λ i / ] k ≤ C (v) (a) kh X − λ i / [ P, X ] h X − λ i − / k ≤ C (b) kh X − λ i / [ P, Y ] h X − λ i − / k ≤ C The key tool for proving these estimates is to use the Riesz formula for the projector P . Since P is a projector onto σ , we can find a contour C of finite length enclosing σ such that:(40) P = 12 πi Z C ( z − H ) − d z and sup z ∈C k ( z − H ) − k < ∞ . To streamline our estimates for P , we will now state two useful lemmas: Lemma B.1.
Suppose that H satisfies Assumption 1 and f, g ∈ C ( R ) . Then we have thefollowing identities: e f He − f = H + 2 i ∇ f · ( i ∇ + A ) − ∆ f + ∇ f · ∇ f (41) e f [ H, g ] e − f = − ∆ g + 2 i ∇ g · ( i ∇ + A ) + 2 ∇ g · ∇ f (42) Proof.
It follows from a direct calculation. (cid:3)
Lemma B.2.
Suppose that H satisfies Assumptions 1 and 2. If z ∈ C , where C is the contour fromEquation (40) , then there exists an absolute constant C such that for all v ∈ R : k v · ( i ∇ + A )( z − H ) − k ≤ C k v k Proof.
Follows from Lemma A.3 in [20]. (cid:3)
With these two lemmas in hand, we will now prove the first three estimates (i), (ii), (iii) inAppendix B.1 and the last two estimates (iv), (v) in Appendix B.2B.1.
Exponential Localization Bounds.
Let us first make use of Lemmas B.1 and B.2 to provethe following corollary
Corollary B.3.
Suppose that H satisfies Assumptions 1 and 2. If z ∈ C , where C is the contourfrom Equation (40) , then there exists constants ( γ ∗ , K , K , K ) independent of z such that for all γ ≤ γ ∗ we have the following bounds:(1) k ( H γ − H )( z − H ) − k ≤ K γ (2) (a) k [ H γ , X ]( z − H ) − k ≤ K (b) k [ H γ , Y ]( z − H ) − k ≤ K (3) (a) k [ H γ − H, X ]( z − H ) − k ≤ K γ (b) k [ H γ − H, Y ]( z − H ) − k ≤ K γ Proof.
We will show how to prove k [ H γ − H, X ]( z − H ) − k ≤ K γ using Lemmas B.1 and B.2. Theremaining estimates follow by similar steps.Choosing f ( x, y ) = γ p x − a ) + ( y − b ) and g ( x, y ) = x with Lemma B.1 we have that[ H γ , X ] = 2 ie · ( i ∇ + A ) + 2 γe · x − a p x − a ) + ( y − a ) ! Similarly, choosing f ( x, y ) ≡ g ( x, y ) = x we have that[ H, X ] = 2 ie · ( i ∇ + A ) . Therefore, [ H γ − H, X ] = 2 γe · x − a p x − a ) + ( y − a ) ! . Therefore, clearly there exists a constant K such that k [ H γ − H, X ]( z − H ) − k ≤ K γ . (cid:3) B.1.1.
Proof of Bound (i).
We calculate P γ − P = 12 πi Z C B γ ( z − H ) − B − γ − ( z − H ) − d z = 12 πi Z C ( z − H γ ) − − ( z − H ) − d z = 12 πi Z C ( z − H γ ) − ( H γ − H )( z − H ) − d z where the final line follows by the second resolvent identity. Hence k P γ − P k ≤ ℓ ( C )2 π sup z ∈C (cid:16) k ( z − H γ ) − kk ( H γ − H )( z − H ) − k (cid:17) The term ( z − H γ ) − is bounded since( z − H γ ) − = ( z − H + H − H γ ) − = ( z − H ) − (cid:16) I − ( H γ − H )( z − H ) − (cid:17) − (43)Due to Corollary B.3(1) we know that k ( H γ − H )( z − H ) − k ≤ K γ. Therefore if we pick γ ≤ (2 K ) − we conclude that k ( z − H γ ) − k ≤ k ( z − H ) − k (cid:16) − (cid:17) − Using this bound, we therefore conclude that k P γ − P k ≤ ℓ ( C )2 π sup z ∈C (cid:16) k ( z − H ) − k (cid:17)(cid:0) K γ (cid:1) which proves bound (i). (cid:3) B.1.2.
Proof of Bound (ii).
We will show this bound for X only, the corresponding bound for Y follows by analogous steps. We calculate[ P γ , X ] = 12 πi Z C [( z − H γ ) − , X ] d z = 12 πi Z C ( z − H γ ) − [ H γ , X ]( z − H γ ) − d z Hence k [ P γ , X ] k ≤ ℓ ( C )2 π sup z ∈C (cid:16) k ( z − H γ ) − kk [ H γ , X ]( z − H γ ) − k (cid:17) . Following the calculation from Equation (43) we have that[ H γ , X ]( z − H γ ) − = [ H γ , X ]( z − H ) − (cid:16) I − ( H γ − H )( z − H ) − (cid:17) − . Therefore, following reasoning from Appendix B.1.1, we conclude that for all γ ≤ (2 K ) − k [ P γ , X ] k ≤ ℓ ( C )2 π sup z ∈C (cid:20)(cid:16) k ( z − H ) − k (cid:17)(cid:16) k [ H γ , X ]( z − H ) − k (cid:17)(cid:21) . LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 35
The above is bounded by a constant due to Corollary B.3(2) which proves bound (ii). (cid:3)
B.1.3.
Proof of Bound (iii).
We will show this bound for X only, the corresponding bound for Y follows by analogous steps. From the calculation in Appendix B.1.1 we have that P γ − P = 12 πi Z C ( z − H γ ) − ( H γ − H )( z − H ) − d z Hence [ P γ − P, X ] = 12 πi Z C [( z − H γ ) − ( H γ − H )( z − H ) − , X ] d z. Now to complete this proof, we need to compute the following commutator[( z − H γ ) − ( H γ − H )( z − H ) − , X ] . This proceeds by an long but straightforward calculation. We start by commuting X with ( z − H ) − :( z − H γ ) − ( H γ − H )( z − H ) − X = (cid:18) ( z − H γ ) − ( H γ − H ) (cid:19)(cid:18) X ( z − H ) − + [( z − H ) − , X ] (cid:19) = (cid:18) ( z − H γ ) − ( H γ − H ) (cid:19)(cid:18) X + ( z − H ) − [ H, X ] (cid:19) ( z − H ) − Next, commuting H γ − H with X gives us that:( z − H γ ) − ( H γ − H ) X ( z − H ) − = ( z − H γ ) − (cid:18) X ( H γ − H ) + [ H γ − H, X ] (cid:19) ( z − H ) − . Finally, commuting ( z − H γ ) − with X gives that:( z − H γ ) − X ( H γ − H )( z − H ) − = (cid:18) X ( z − H γ ) − + [( z − H γ ) − , X ] (cid:19) ( H γ − H )( z − H ) − = (cid:18) X + ( z − H γ ) − [ H γ , X ] (cid:19) ( z − H γ ) − ( H γ − H )( z − H ) − . Combining all of these estimates together shows that:[( z − H γ ) − ( H γ − H )( z − H ) − , X ]= ( z − H γ ) − ( H γ − H )( z − H ) − [ H, X ]( z − H ) − + ( z − H γ ) − [ H γ − H, X ]( z − H ) − + ( z − H γ ) − [ H γ , X ]( z − H γ ) − ( H γ − H )( z − H ) − . Hence k [( z − H γ ) − ( H γ − H )( z − H ) − , X ] k≤ (cid:16) k ( z − H γ ) − k (cid:17)(cid:16) k ( H γ − H )( z − H ) − k (cid:17)(cid:16) k [ H, X ]( z − H ) − k (cid:17) + (cid:16) k ( z − H γ ) − k (cid:17)(cid:16) k [ H γ − H, X ]( z − H ) − k (cid:17) + (cid:16) k ( z − H γ ) − k (cid:17)(cid:16) k [ H γ , X ]( z − H γ ) − k (cid:17)(cid:16) k ( H γ − H )( z − H ) − k (cid:17) . We showed in Appendix B.1.1 that k ( z − H γ ) − k is bounded and in Appendix B.1.2 that k [ H γ , X ]( z − H γ ) − k is bounded. These bounds combined with Corollary B.3 implies bound (iii). (cid:3) B.2.
Square Root Localization Bounds.
Similar to Appendix B.1 we will begin this section bystating a corollary which follows from Lemmas B.1 and B.2.
Corollary B.4.
Suppose that H satisfies Assumptions 1 and 2. If z ∈ C , where C is the contourfrom Equation (40) , then there exists constants ( K , K ) independent of z such that for all λ ∈ R :(1) k [ H, h X − λ i / ]( z − H ) − k ≤ K (2) (a) kh X − λ i / [ H, X ] h X − λ i − / ( z − H ) − k ≤ K (b) kh X − λ i / [ H, Y ] h X − λ i − / ( z − H ) − k ≤ K Proof.
Applying Lemma B.1 with f ( x, y ) ≡ g ( x, y ) = h x − λ i / gives[ H, h X − λ i / ] = (cid:18) − ( x − λ ) h x − λ i / (cid:19) + (cid:18) i ( x − λ )2 h x − λ i / (cid:19) e · ( i ∇ + A ) . Hence, [ H, h X − λ i / ]( z − H ) − is clearly bounded by a constant due to Lemma B.2.To get the second and third bounds, simply apply Lemma B.1 with f ( x, y ) = ln ( h x − λ i / ) andeither g ( x, y ) = x or g ( x, y ) = y and use a similar argument. (cid:3) B.2.1.
Proof of Bound (iv).
We calculate[ P, h X − λ i / ] = 12 πi Z C [( z − H ) − , h X − λ i / ] d z = 12 πi Z C ( z − H ) − [ H, h X − λ i / ]( z − H ) − d z. Hence k [ P, h X − λ i / ] k ≤ ℓ ( C )2 π sup z ∈C (cid:16) k ( z − H ) − kk [ H, h X − λ i / ]( z − H ) − k (cid:17) which is bounded by a constant due to Corollary B.4. (cid:3) B.2.2.
Proof of Bound (v).
We will only prove the bound for Y , the bound for X follows by similarsteps. We calculate [ P, Y ] = 12 πi Z C [( z − H ) − , Y ] d z = 12 πi Z C ( z − H ) − [ H, Y ]( z − H ) − d z. Therefore, kh X − λ i / [ P, Y ] h X − λ i − / k≤ ℓ ( C )2 π sup z ∈C (cid:16) kh X − λ i / ( z − H ) − [ H, Y ]( z − H ) − h X − λ i − / k (cid:17) To help with reducing clutter in the next few steps, let us define the shorthand X λ := X − λ . Withthis notation, we have that kh X − λ i / [ P, Y ] h X − λ i − / k≤ ℓ ( C )2 π sup z ∈C kh X λ i / ( z − H ) − [ H, Y ]( z − H ) − h X λ i − / k≤ ℓ ( C )2 π sup z ∈C (cid:16) kh X λ i / ( z − H ) − h X λ i − / kkh X λ i / [ H, Y ]( z − H ) − h X λ i − / k (cid:17) The first term on the right-hand side in the bracket is clearly bounded since kh X λ i / ( z − H ) − h X λ i − / k LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 37 = k [ h X λ i / , ( z − H ) − ] h X λ i − / + ( z − H ) − k≤ k ( z − H ) − [ h X λ i / , H ]( z − H ) − h X λ i − / k + k ( z − H ) − k≤ k ( z − H ) − k k [ h X λ i / , H ]( z − H ) − k + k ( z − H ) − k which is bounded due to Corollary B.4(1).To see that h X λ i / [ H, Y ]( z − H ) − h X λ i − / is bounded we perform the following calculations h X λ i / [ H, Y ]( z − H ) − h X λ i − / = h X λ i / [ H, Y ] h X λ i − / h X λ i / ( z − H ) − h X λ i − / = h X λ i / [ H, Y ] h X λ i − / (cid:16) [ h X λ i / , ( z − H ) − ] h X λ i − / + ( z − H ) − (cid:17) = h X λ i / [ H, Y ] h X λ i − / ( z − H ) − (cid:16) [ h X λ i / , H ]( z − H ) − h X λ i − / + I (cid:17) Hence kh X λ i / [ H, Y ]( z − H ) − h X λ i − / k≤ (cid:16) kh X λ i / [ H, Y ] h X λ i − / ( z − H ) − k (cid:17)(cid:16) k [ h X λ i / , H ]( z − H ) − kkh X λ i − / k + 1 (cid:17) Since [ h X − λ i / , H ]( z − H ) − , and h X − λ i / [ H, Y ] h X − λ i − / ( z − H ) − are bounded due to Corollary B.4, we conclude bound (v). (cid:3) Appendix C. Technical Lemmas
We collect two technical lemmas here which will be used in other parts of the proof.C.1.
Decay Lemma.
Recall, for each ( k, ℓ ) ∈ Z we define the unit box centered at ( k, ℓ ) S k,ℓ := (cid:20) k − , k + 12 (cid:19) × (cid:20) ℓ − , ℓ + 12 (cid:19) Also recall our special notation for the characteristic function for S k,ℓ χ k,ℓ ( x ) = ( x ∈ S k,ℓ x S k,ℓ Using this notation, we now state the following result:
Lemma C.1.
For any s , s ≥ , any ( m, n ) ∈ Z , any ( k, ℓ ) ∈ Z , and any v ∈ L ( R ) then (44) k χ k,ℓ v k ≤ s + s k χ k,ℓ ( | X − m | + 1) s ( | Y − n | + 1) s v kh m − k i s h n − ℓ i s where h x i is the Japanese bracket h x i := √ x .Proof. Instead of proving Equation (44) directly we will instead prove that:(45) k χ k,ℓ v k ≤ k χ k,ℓ ( | X − m | + 1) s ( | Y − n | + 1) s v k|| m − k | + 1 / | s || n − ℓ | + 1 / | s Proving Equation (45) is sufficient since for all a ∈ Z one can check that p a ≤ || a | + 1 / | . Therefore, for any ( m, n ) ∈ Z and any ( k, ℓ ) ∈ Z we have that || m − k | + 1 / | − s ≤ s h m − k i − s || n − ℓ | + 1 / | − s ≤ s h n − ℓ i − s . Hence, the proving Equation (45) implies Equation (44).We will now prove Equation (45) in the case where m = k and n = ℓ ; the other cases followeasily using similar arguments. Our main tool for proving Equation (45) will be to use “strip”characteristic functions in X and Y : χ {| x − m | ≤ d } ( x ) = ( | x − m | ≤ d .χ {| x − n | ≤ d } ( x ) = ( | x − n | ≤ d . The key observation is that characteristic functions χ k,ℓ and χ {| x − m | ≤ | m − k | − / } have disjoint supports (up to a set of measure zero). Therefore,(46) χ k,ℓ = χ k,ℓ (1 − χ {| x − m | ≤ | m − k | − / } ) . Using Equation (46) for any function v we have that: k χ k,ℓ v k = Z R χ k,ℓ | v ( x ) | d x = Z R χ k,ℓ (1 − χ {| x − m | ≤ | m − k | − / } ) | v ( x ) | d x = Z R χ k,ℓ (1 − χ {| x − m | ≤ | m − k | − / } ) (1 + | x − m | ) s (1 + | x − m | ) s | v ( x ) | d x Since 1 − χ {| x − m | ≤ | m − k | − / } = χ {| x − m | > | m − k | − / } we have(1 − χ {| x − m | ≤ | m − k | − / } ) 1(1 + | x − m | ) s = χ {| x − m | > | m − k | − / } | x − m | ) s ≤ | m − k | + 1 / s Hence k χ k,ℓ v k ≤ | m − k | + 1 / s Z R χ k,ℓ (1 + | x − m | ) s | v ( x ) | d x = k χ k,ℓ (1 + | X − m | ) s v k ( | m − k | + 1 / s We will now apply a similar argument k χ k,ℓ (1 + | X − m | ) s v k . By similar reasoning to Equation(46), we have that(47) χ k,ℓ = χ k,ℓ (1 − χ {| x − n | ≤ | n − ℓ | − / } ) . LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 39
Therefore, k χ k,ℓ (1 + | X − m | ) s v k = Z R χ k,ℓ (1 + | x − m | ) s | v ( x ) | d x = Z R χ k,ℓ (1 − χ {| x − n | ≤ | n − ℓ | − / } )(1 + | x − m | ) s | v ( x ) | d x = Z R χ k,ℓ (1 − χ {| x − n | ≤ | n − ℓ | − / } ) (1 + | x − m | ) s (1 + | x − n | ) s (1 + | x − n | ) s | v ( x ) | d x . Hence, repeating a similar argument to before we conclude that: k χ k,ℓ v k ≤ k χ k,ℓ (1 + | X − m | ) s (1 + | Y − n | ) s v k ( | m − k | + 1 / s ( | n − ℓ | + 1 / s . This proves Equation (45) proving the lemma. (cid:3)
C.2.
Product to Sum Bound.Lemma C.2.
For any s , s ≥ , any ( m, n ) ∈ R , and any v ∈ L ( R ) we have the followinginequality: k (1+ | X − m | ) s (1 + | Y − n | ) s v k≤ k (1 + | X − m | ) s + s v k + k (1 + | Y − n | ) s + s v k . Proof.
Observe that the result is trivial if s = 0 or s = 0 so we can assume without loss ofgenerality that s > s > k (1+ | X − m | ) s (1 + | Y − n | ) s v k = Z R (1 + | x − m | ) s (1 + | x − n | ) s | v ( x ) | d x . Since s , s > p = s + s s and q = s + s s so that(1+ | x − m | ) s (1 + | x − n | ) s ≤ p (1 + | x − m | ) s p + 1 q (1 + | x − n | ) s q ≤ p (1 + | x − m | ) s + s ) + 1 q (1 + | x − n | ) s + s ) Hence, using this pointwise bound: k (1 + | X − m | ) s (1 + | Y − n | ) s v k ≤ p k (1 + | X − m | ) s + s v k + 1 q k (1 + | Y − n | ) s + s v k The result follows by taking square roots, using that √ a + b ≤ | a | + | b | , and observing thatmax { p − / , q − / } ≤ (cid:3) Appendix D. Proof of Proposition 6.2
We’ll start this section by recalling the proposition we want to prove:
Proposition 6.2.
Fix an orthonormal basis { ψ ( j ) m,n } . For any h, g ∈ L ( R ) define h m,n,j and g m ′ ,n ′ ,j ′ as follows: h m,n,j := Z R | ψ ( j ) m,n ( x ) h ( x ) | d x g m ′ ,n ′ ,j ′ := Z R | ψ ( j ′ ) m ′ ,n ′ ( x ) g ( x ) | d x . If { ψ ( j ) m,n } is an s -localized basis with s > then there exists an absolute constant C such that X m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| h m,n,j g m ′ ,n ′ ,j ′ ≤ C k h kk g k The core idea of this estimate is to make use of a modified version of Young’s convolution in-equality. Let’s start by recalling the standard Young’s convolution inequality for three functions [9,Theorem 4.2]:
Theorem 3 (Young’s Convolution Inequality, Discrete Case) . Suppose that g, h, k are functions sothat g ∈ ℓ p ( Z ) , h ∈ ℓ q ( Z ) , k ∈ ℓ r ( Z ) where p, q, r ≥ and p + 1 q + 1 r = 2 . Then we have the following bound: X m,n X m ′ ,n ′ | g [ m ′ , n ′ ] k [ m − m ′ , n − n ′ ] h [ m, n ] | ≤ k g k ℓ p k k k ℓ r k h k ℓ q Roughly speaking, one can think of the proof of the proposition as applying this inequality with p = q = 2 and r = 1 where h = h m,n,j g = g m ′ ,n ′ ,j ′ k = |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| . It is important to note that |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| is not a convolution kernel so Young’s convo-lution inequality does not directly apply. Despite this, some minor modifications to the proof ofYoung’s convolution inequality let us derive a similar result for our specific case.With this argument in mind, in the following sections we prove the following two lemmas: Lemma D.1. If h ∈ L ( R ) and the collection { ψ ( j ) m,n } is an s -localized orthonormal basis for s > with finitely degenerate centers then there exists a finite constant C > such that X m,n,j h m,n,j = X m,n,j Z R | ψ ( j ) m,n ( x ) h ( x ) | d x ! ≤ C k h k . Lemma D.2.
If the collection { ψ ( j ) m,n } is an s -localized orthonormal basis for s > with finitelydegenerate centers then there exists a finite constant C ′ > such that: sup m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| ≤ C ′ sup m ′ ,n ′ ,j ′ X m,n,j |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| ≤ C ′ LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 41
Assuming these lemmas are true, we can now complete the proof of Proposition 6.2. First, byapplying Cauchy-Schwarz to the sum over ( m, n, j ) and ( m ′ , n ′ , j ′ ) we get: X m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| h m,n,j g m ′ ,n ′ ,j ′ = X m,n,j X m ′ ,n ′ ,j ′ (cid:16) |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| / h m,n,j (cid:17)(cid:16) |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| / g m ′ ,n ′ ,j ′ (cid:17) ≤ (cid:18) X m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| h m,n,j (cid:19) / × (cid:18) X m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| g m ′ ,n ′ ,j ′ (cid:19) / Next, since all of the terms in the summand are positive, we can take the supremum over ( m, n, j )in the first sum and the supremum over ( m ′ , n ′ , j ′ ) in the second sum to get: X m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| h m,n,j ≤ (cid:18) sup m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| (cid:19)(cid:18) X m,n,j h m,n,j (cid:19)X m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| g m ′ ,n ′ ,j ′ ≤ (cid:18) sup m ′ ,n ′ ,j ′ X m,n,j |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| (cid:19)(cid:18) X m ′ ,n ′ ,j ′ g m ′ ,n ′ ,j ′ (cid:19) Hence applying Lemma D.1 and Lemma D.2 we conclude that there exists a constant C X m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| h m,n,j g m ′ ,n ′ ,j ′ ≤ C k h kk g k proving the proposition.We now turn to prove Lemmas D.1 and D.2. Our main technique for proving these results is toinsert a partition of unity of the form: X k,ℓ χ k,ℓ ( x ) = 1where χ k,ℓ ( x ) = ( x ∈ (cid:2) k − , k + (cid:1) × (cid:2) ℓ − , ℓ + (cid:1) . We prove two technical lemmas (Lemmas C.1 and C.2) which relate the characteristic functions χ k,ℓ to the basis { ψ ( j ) m,n } in Appendix C.D.1. Proof of Lemma D.1.
Before starting this proof, we will recall an alternate statement ofYoung’s convolution inequality which follows from a duality argument applied to Theorem 3:
Theorem 4 (Young’s Convolution Inequality, Discrete Case, Alternative Statement) . Suppose that g, k are functions so that g ∈ ℓ p ( Z ) , k ∈ ℓ q ( Z ) where p, q, r ≥ and p + 1 q = 1 + 1 r . Then we have the following bound: X m,n X m ′ ,n ′ | g [ m ′ , n ′ ] k [ m − m ′ , n − n ′ ] | r ≤ k g k rℓ p k k k rℓ q Inserting the partition of unity P k,ℓ χ k,ℓ = 1 to the quantity we want to bound gives: X m,n,j (cid:18)Z R | ψ ( j ) m,n ( x ) h ( x ) | d x (cid:19) = X m,n,j X k,ℓ Z R | χ k,ℓ ( x ) ψ ( j ) m,n ( x ) h ( x ) | d x ≤ X m,n,j X k,ℓ k χ k,ℓ ψ ( j ) m,n kk χ k,ℓ h k ≤ M sup j X m,n X k,ℓ k χ k,ℓ ψ ( j ) m,n kk χ k,ℓ h k (48)where in the second to last line we have used the Cauchy-Schwarz inequality. The key observationhere is to note that the sequence {k χ k,ℓ h k} ( k,ℓ ) ∈ Z is square summable. In particular, we have that X k,ℓ k χ k,ℓ h k = X k,ℓ Z χ k,ℓ | h ( x ) | d x = Z | h ( x ) | d x = k h k . By applying Lemma C.1 with s = s = 1 + ǫ and Lemma C.2 we have for any ǫ > k χ k,ℓ ψ ( j ) m,n k ≤ C k ( | X − m | + 1) ǫ ( | Y − n | + 1) ǫ ψ ( j ) m,n kh m − k i ǫ h n − ℓ i ǫ ≤ C ( k ( | X − m | + 1) ǫ ψ ( j ) m,n k + k ( | Y − n | + 1) ǫ ψ ( j ) m,n k ) h m − k i ǫ h n − ℓ i ǫ Hence, assuming ψ ( j ) m,n is s -localized with s >
2, we can pick ǫ sufficiently small so that k χ k,ℓ ψ ( j ) m,n k ≤ C h m − k i ǫ h n − ℓ i ǫ . Substituting this inequality into Equation (48) the sum to bound is therefore:(49) X m,n X k,ℓ k χ k,ℓ h kh m − k i ǫ h n − ℓ i ǫ where we have dropped the supremum over j since the summand no longer depends on j . Nownotice that Equation (49) is the ℓ norm squared of a convolution. Since k χ k,ℓ h k ∈ ℓ ( Z ) and 1 h m i ǫ h n i ǫ ∈ ℓ ( Z )applying Young’s convolution inequality with p = r = 2 , and q = 1, we conclude that X m,n X k,ℓ k χ k,ℓ h kh m − k i ǫ h n − ℓ i ǫ ≤ C k h k and the lemma is proved. LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 43
D.2.
Proof of Lemma D.2.
Let us begin this proof by recalling the bounds we want to showsup m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| ≤ C ′ sup m ′ ,n ′ ,j ′ X m,n,j |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| ≤ C ′ . Our first observation is that these two bounds are in fact equivalent. Since { ψ ( j ) m,n } forms anorthonormal basis, we have that whenever ( m, n, j ) = ( m ′ , n ′ , j ′ )(50) h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i = h ψ ( j ) m,n , ( X − m ′ ) ψ ( j ′ ) m ′ ,n ′ i Since Equation (50) is also true when ( m, n, j ) = ( m ′ , n ′ , j ′ ), it follows that for any fixed choice of( m ′ , n ′ , j ′ ) we have X m,n,j |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| = X m,n,j |h ψ ( j ) m,n , ( X − m ′ ) ψ ( j ′ ) m ′ ,n ′ i| Therefore, the bounds we want to show aresup m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| ≤ C ′ sup m ′ ,n ′ ,j ′ X m,n,j |h ψ ( j ) m,n , ( X − m ′ ) ψ ( j ′ ) m ′ ,n ′ i| ≤ C ′ . Using the fact that X is self-adjoint we easily see that the second bound is equivalent to the firstbound by making the relabeling ( m, n, j ) ⇔ ( m ′ , n ′ , j ′ ).For the rest of this argument, let us take ( m, n, j ) to be fixed and prove a bound which isindependent of the choice of ( m, n, j ). Inserting a partition of unity of characteristic functions P k,ℓ χ k,ℓ = 1 we have that: X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i|≤ X m ′ ,n ′ ,j ′ X k,ℓ |h χ k,ℓ ( X − m ) ψ ( j ) m,n , ψ ( j ′ ) m ′ ,n ′ i|≤ X m ′ ,n ′ ,j ′ X k,ℓ k χ k,ℓ ( X − m ) ψ ( j ) m,n kk χ kℓ ψ ( j ′ ) m ′ ,n ′ k≤ M sup j ′ X m ′ ,n ′ X k,ℓ k χ k,ℓ ( X − m ) ψ ( j ) m,n kk χ kℓ ψ ( j ′ ) m ′ ,n ′ k ≤ M sup j ′ X m ′ ,n ′ X k,ℓ k χ k,ℓ ( | X − m | + 1) ψ ( j ) m,n kk χ kℓ ψ ( j ′ ) m ′ ,n ′ k (51) where in the last line we have used the pointwise bound ( x − m ) ≤ ( | x − m | + 1) . ApplyingLemma C.1 with s = s = 1 + ǫ gives us that: k χ k,ℓ ψ ( j ′ ) m ′ ,n ′ k ≤ C k χ k,ℓ ( | X − m ′ | + 1) ǫ ( | Y − n ′ | + 1) ǫ ψ ( j ′ ) m ′ ,n ′ kh m ′ − k i ǫ h n ′ − ℓ i ǫ (52)Next, applying Lemma C.1 with s = s = 1 / ǫ gives that: k χ k,ℓ ( | X − m | + 1) ψ ( j ) m,n k≤ C k χ k,ℓ ( | X − m | + 1) / ǫ ( | Y − n | + 1) / ǫ ψ ( j ) m,n kh m − k i / ǫ h n − ℓ i / ǫ . (53)Applying Lemma C.2 to upper bound Equation (52) gives: k χ kℓ ψ ( j ′ ) m ′ ,n ′ k ≤ C k χ k,ℓ ( | X − m ′ | + 1) ǫ ( | Y − n ′ | + 1) ǫ ψ ( j ′ ) m ′ ,n ′ kh m ′ − k i ǫ h n ′ − ℓ i ǫ ≤ C ( k ( | X − m ′ | + 1) ǫ ψ ( j ′ ) m ′ ,n ′ k + k ( | Y − n ′ | + 1) ǫ ψ ( j ′ ) m ′ ,n ′ k ) h m ′ − k i ǫ h n ′ − ℓ i ǫ ≤ C h m ′ − k i ǫ h n ′ − ℓ i ǫ . (54)In the last line, we have used the fact that by assumption { ψ ( j ) m,n } is s -localized with s >
2, so wecan pick ǫ sufficiently small so that k ( | X − m ′ | + 1) ǫ ψ ( j ′ ) m ′ ,n ′ k + k ( | Y − n ′ | + 1) ǫ ψ ( j ′ ) m ′ ,n ′ k is bounded by a constant.Plugging Equation (53) and Equation (54) into Equation (51) we have that we can find a constant C such that: sup j ′ X m ′ ,n ′ X k,ℓ k χ k,ℓ ( | X − m | + 1) ψ ( j ) m,n kk χ kℓ ψ ( j ′ ) m ′ ,n ′ k≤ X m ′ ,n ′ X k,ℓ C k χ k,ℓ ( | X − m | + 1) / ǫ ( | Y − n | + 1) / ǫ ψ ( j ) m,n kh m − k i / ǫ h n − ℓ i / ǫ h m ′ − k i ǫ h n ′ − ℓ i ǫ . Treating ( m, n, j ) as a constant we can group the summand into two parts A k,ℓ := k χ k,ℓ ( | X − m | + 1) / ǫ ( | Y − n | + 1) / ǫ ψ ( j ) m,n kh m − k i / ǫ h n − ℓ i / ǫ B m ′ − k,n ′ − ℓ := 1 h m ′ − k i ǫ h n ′ − ℓ i ǫ . With this notation the sum we wish to bound can be written as:(55) X m ′ ,n ′ X k,ℓ A k,ℓ B m ′ − k,n ′ − ℓ . However the above sum is the ℓ -norm of a convolution. Therefore, if we can show that A k,ℓ ∈ ℓ ( Z ) and B m ′ ,n ′ ∈ ℓ ( Z ) then by applying Young’s convolution inequality (Theorem 4) with p = q = r = 1 we can conclude that Equation (55) is bounded, completing the proof of Lemma D.2. LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 45
It’s easy to see that B m ′ ,n ′ ∈ ℓ ( Z ) for any ǫ > A k,ℓ ∈ ℓ ( Z ). Applying Cauchy-Schwarz inequality X k,ℓ A k,ℓ = X k,ℓ k χ k,ℓ ( | X − m | + 1) / ǫ ( | Y − n | + 1) / ǫ ψ ( j ) m,n kh m − k i / ǫ h n − ℓ i / ǫ ≤ X k,ℓ k χ k,ℓ ( | X − m | + 1) / ǫ ( | Y − n | + 1) / ǫ ψ ( j ) m,n k / × X k,ℓ h m − k i ǫ h n − ℓ i ǫ / ≤ C k ( | X − m | + 1) / ǫ ( | Y − n | + 1) / ǫ ψ ( j ) m,n k where in the last line we have made use of the fact that P k,ℓ χ k,ℓ = 1. Since by assumption { ψ ( j ) m,n } is s -localized with s >
2, by applying Lemma C.2 we conclude that k ( | X − m | + 1) / ǫ ( | Y − n | + 1) / ǫ ψ ( j ) m,n k is bounded for all ǫ sufficiently small. This completes the proof of Lemma D.2. Appendix E. Square Root Bounds
In this section, we will prove the following lemma which includes Lemma 8.3 as a special case.
Lemma E.1.
Suppose that P is a projector satisfying decay estimates as Assumption 4. Supposefurther that P admits a basis with finite degenerate centers which is s -localized for some s > .Then there exists a constant C > such that for any λ ∈ G (recall that G is the set of gaps definedin Equation (23) ): k S λ P h X − λ i / k ≤ C kh X − λ i / P S λ k ≤ C k S − λ P h X − λ i − / k ≤ C kh X − λ i − / P S − λ k ≤ C Lemma E.1 follows as an easy corollary of the following result and our decay estimates on P (Assumption 4 and Corollary 5.1): Lemma E.2.
Suppose that P is a projector satisfying decay estimates as Assumption 4. Supposefurther that P admits a basis with finite degenerate centers which is s -localized for some s > .Then there exists a constant C ′ > such that for any λ ∈ G : k P S − λ P − P h X − λ i / P k ≤ C ′ Let’s assume Lemma E.2 is true and prove Lemma E.1. We will return to prove Lemma E.2 inthe next section (Appendix E.1).
Proof of Lemma E.1.
We will show that k S λ P h X − λ i / k ≤ C k S − λ P h X − λ i − / k ≤ C the other two bounds follow by using the fact for any bounded operator k A k = k A † k . For the firstbound, we calculate S λ P h X − λ i / = S λ P (cid:16) h X − λ i / − S − λ + S − λ (cid:17) = S λ P (cid:16) h X − λ i / − S − λ (cid:17) + P = S λ P (cid:16) h X − λ i / − S − λ (cid:17) ( P + Q ) + P = S λ P (cid:16) h X − λ i / − S − λ (cid:17) P + S λ P h X − λ i / Q + P, where we have used that [ P, S λ ] = 0, P + Q = I , and P Q = QP = 0. Therefore, we have that k S λ P h X − λ i / k≤ k S λ k (cid:16) k P h X − λ i / P − P S − λ P k + k P h X − λ i / Q k (cid:17) + 1 ≤ k S λ k (cid:16) k P h X − λ i / P − P S − λ P k + k [ P, h X − λ i / ] k (cid:17) + 1 . Now observe that k P h X − λ i / P − P S − λ P k is bounded due to Lemma E.2 and k [ P, h X − λ i / ] k is bounded due to decay estimates on P (Assumption 4(iv)). Hence the first bound is proved.For the second bound, we calculate S − λ P h X − λ i − / = (cid:16) S − λ − h X − λ i / + h X − λ i / (cid:17) P h X − λ i − / = (cid:16) S − λ − h X − λ i / (cid:17) P h X − λ i − / + h X − λ i / P h X − λ i − / = (cid:16) S − λ − h X − λ i / (cid:17) P h X − λ i − / + [ h X − λ i / , P ] h X − λ i − / + P Hence, using the fact that kh X − λ i − / k ≤
1, we get the upper bound k S − λ P h X − λ i − / k≤ k (cid:16) S − λ − h X − λ i / (cid:17) P k + k [ h X − λ i / , P ] k + 1 ≤ k P (cid:16) S − λ − h X − λ i / (cid:17) P k + k Q h X − λ i / P k + k [ h X − λ i / , P ] k + 1 ≤ k P (cid:16) S − λ − h X − λ i / (cid:17) P k + 2 k [ h X − λ i / , P ] k + 1which is bounded due to Lemma E.2 and Assumption 4(iv) as before. (cid:3) E.1.
Proof of Lemma E.2.
The proof of this lemma follows very closely with the proof ofLemma D.2 from Appendix D. Writing out these two expressions in terms of the basis { ψ ( j ) m,n } we have that: P S − λ P = X m,n,j | m − λ | / | ψ ( j ) m,n ih ψ ( j ) m,n | P h X − λ i / P = X m,n,j X m ′ ,n ′ ,j ′ h ψ ( j ) m,n , h X − λ i / ψ ( j ′ ) m ′ ,n ′ i| ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ | Since { ψ ( j ) m,n } is orthonormal, we have that when ( m, n, j ) = ( m ′ , n ′ , j ′ ): h ψ ( j ) m,n , h X − λ i / ψ ( j ′ ) m ′ ,n ′ i = h ψ ( j ) m,n , (cid:16) h X − λ i / − | m − λ | / (cid:17) ψ ( j ′ ) m ′ ,n ′ i = h ψ ( j ) m,n , (cid:16) h X − λ i / − | m ′ − λ | / (cid:17) ψ ( j ′ ) m ′ ,n ′ i (56) LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 47
Therefore, we can express the difference
P S − λ P − P h X − λ i / P as follows: P S − λ P − P h X − λ i / P = − X m,n,j X m ′ ,n ′ ,j ′ h ψ ( j ) m,n , (cid:16) h X − λ i / − | m ′ − λ | / (cid:17) ψ ( j ′ ) m ′ ,n ′ i| ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ | Now by Schur’s test, to show that
P S − λ P − P h X − λ i / P is a bounded operator, it’s enough toshow that: sup m,n,j X m ′ ,n ′ ,j ′ |h ψ ( j ) m,n , (cid:16) h X − λ i / − | m − λ | / (cid:17) ψ ( j ′ ) m ′ ,n ′ i| < ∞ sup m ′ ,n ′ ,j ′ X m,n,j |h ψ ( j ) m,n , (cid:16) h X − λ i / − | m ′ − λ | / (cid:17) ψ ( j ′ ) m ′ ,n ′ i| < ∞ Due to Equation (56) and the fact that X is self-adjoint, we can see that the above two boundsare equivalent so it is enough to prove the first of these bounds. Inserting the partition of unity P k,ℓ χ k,ℓ = 1 we have that X m ′ ,n ′ ,j ′ X k,ℓ |h χ k,ℓ ψ ( j ) m,n , (cid:16) h X − λ i / − | m − λ | / (cid:17) ψ ( j ′ ) m ′ ,n ′ i|≤ X m ′ ,n ′ ,j ′ X k,ℓ k χ k,ℓ (cid:16) h X − λ i / − | m − λ | / (cid:17) ψ ( j ) m,n kk χ k,ℓ ψ ( j ′ ) m ′ ,n ′ k≤ X m ′ ,n ′ ,j ′ X k,ℓ k χ k,ℓ ( | X − m | + 1) / ψ ( j ) m,n kk χ k,ℓ ψ ( j ′ ) m ′ ,n ′ k . To get the last line we have used the following sequence of pointwise inequalities: |h x − λ i / − | m − λ | / | ≤ |h x − λ i − | m − λ || (elementary)= | p ( x − λ ) + 1 − p ( m − λ ) + 0 |≤ p ( x − m ) + 1 (reverse triangle inequality) ≤ | x − m | + 1 . (elementary)Therefore, the quantity we want to bound is X m ′ ,n ′ ,j ′ X k,ℓ k χ k,ℓ ( | X − m | + 1) / ψ ( j ) m,n kk χ k,ℓ ψ ( j ′ ) m ′ ,n ′ k≤ M sup j ′ X m ′ ,n ′ X k,ℓ k χ k,ℓ ( | X − m | + 1) / ψ ( j ) m,n kk χ k,ℓ ψ ( j ′ ) m ′ ,n ′ k (57)In the proof of Lemma D.2, we showed that the following expression is bounded when P admits abasis which is s -localized for s > j ′ X m ′ ,n ′ X k,ℓ k χ k,ℓ ( | X − m | + 1) ψ ( j ) m,n kk χ k,ℓ ψ ( j ′ ) m ′ ,n ′ k . Since ( | X − m | + 1) / ≤ ( | X − m | + 1) this calculation implies that Equation (57) is bounded,completing the proof of Lemma 8.3. Appendix F. Proof of Proposition 8.4
Let us start this section by recalling the proposition we want to prove:
Proposition 8.4.
Suppose that P is a projector satisfying decay estimates as Assumption 4. Sup-pose further that P admits a basis with finitely degenerate centers which is s -localized for some s > / . Then for any λ ∈ G there exists a finite constant C > such that kh X − λ i − / [ X, ˜ X ] h X − λ i − / k ≤ C, kh X − λ i − / [ Y, ˜ X ] h X − λ i − / k ≤ C. (58)The core idea of this proof is to rewrite the commutators we are interested in bounding intodifferent parts we can control.Let’s begin by considering the commutator [ Y, ˜ X ]. Using that Y = P Y P + P Y Q + QY P + QY Q and ˜ X = P ˜ XP + QXQ we have:[ Y, ˜ X ] = [ P Y P + P Y Q + QY P + QY Q, P ˜ XP + QXQ ]= [
P Y P + P Y Q + QY P, P ˜ XP ] + [ P Y Q + QY P + QY Q, QXQ ]where we have used that
P Q = QP = 0. Grouping the terms with P Y Q + QY P together thengives us: [ Y, ˜ X ] = [ P Y P, P ˜ XP ] + [ QY Q, QXQ ] + [
P Y Q + QY P, ˜ X ]Performing similar calculations for [ X, ˜ X ] gives us[ X, ˜ X ] = [ P XP, P ˜ XP ] + [ QXQ, QXQ ] + [
P XQ + QXP, ˜ X ]= [ P XP, P ˜ XP ] + [ P XQ + QXP, ˜ X ]Therefore, we have three types of terms to bound:(1) [ QY Q, QXQ ] (see Appendix F.1)(2) [
P XQ + QXP, ˜ X ] and [ P Y Q + QY P, ˜ X ] (see Appendix F.2)(3) [ P XP, P ˜ XP ] and [ P Y P, P ˜ XP ] (see Appendix F.3)While [ QY Q, QXQ ] can be bounded without using the decay terms, h X − λ i − / (see Appendix F.1),bounding the other terms requires making use of this additional decay (see Appendix F.2 andAppendix F.3).F.1. Bounding [ QXQ, QY Q ] term. Using the fact that Q = I − P and [ X, Y ] = 0 we easilycalculate that [
QXQ, QY Q ] =
QXQY Q − QY QXQ = QX ( I − P ) Y Q − QY ( I − P ) XQ = QXY Q − QXP Y Q − QY XQ + QY P XQ = QY P XQ − QXP Y Q
Therefore, k [ QXQ, QY Q ] k = k QY P XQ − QXP Y Q k≤ k
QY P kk P XQ k + k QXP kk P Y Q k but this is bounded due to Corollary 5.1(i) with γ = 0. LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 49
F.2.
Bounding [ P XQ + QXP, ˜ X ] and [ P Y Q + QY P, ˜ X ] terms. In this section, we will showhow to bound h X − λ i − / [ P Y Q, ˜ X ] h X − λ i − / . Bounding h X − λ i − / [ QY P, ˜ X ] h X − λ i − / and the other two terms follows using similar steps.Expanding this commutator gives us[ P Y Q, ˜ X ] = [ P Y Q, ˜ X − λ ]= P Y Q ( ˜ X − λ ) − ( ˜ X − λ ) P Y Q.
Let’s start by considering the term h X − λ i − / P Y Q ( ˜ X − λ ) h X − λ i − / . Inserting a copy of h X − λ i / h X − λ i − / gives us that h X − λ i − / P Y Q ( ˜ X − λ ) h X − λ i − / = (cid:16) h X − λ i − / P Y Q h X − λ i / (cid:17)(cid:16) h X − λ i − / ( ˜ X − λ ) h X − λ i − / (cid:17) due to Corollary 5.1(v), we know that h X − λ i − / P Y Q h X − λ i / is bounded. The second termcan be seen to be bounded by adding and subtracting X : kh X − λ i − / ( ˜ X − λ ) h X − λ i − / k = kh X − λ i − / ( ˜ X − X + X − λ ) h X − λ i − / k≤ k ˜ X − X k + k ( X − λ ) h X − λ i − k≤ k ˜ X − X k + 1 . Hence, these terms are bounded.F.3.
Bounding [ P XP, P ˜ XP ] and [ P Y P, P ˜ XP ] terms. In this section we will show how to boundthe following quantities h X − λ i − / [ P XP, P ˜ X P ] h X − λ i − / h X − λ i − / [ P Y P, P ˜ XP ] h X − λ i − / . We’ll start by writing the commutators [
P XP, P ˜ XP ] and [ P Y P, P ˜ XP ] in terms of the basis { ψ ( j ) m,n } .[ P XP,P ˜ XP ] = X m,n,j X m ′ ,n ′ ,j ′ h ψ ( j ) m,n , Xψ ( j ′ ) m ′ ,n ′ i| ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ | , P ˜ XP = X m,n,j X m ′ ,n ′ ,j ′ h ψ ( j ) m,n , Xψ ( j ′ ) m ′ ,n ′ i (cid:16) P ˜ XP | ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ | − | ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ | P ˜ XP (cid:17) = X m,n,j X m ′ ,n ′ ,j ′ ( m − m ′ ) h ψ ( j ) m,n , Xψ ( j ′ ) m ′ ,n ′ i| ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ | (59)A similar calculation shows that(60) [ P Y P, P ˜ XP ] = X m,n,j X m ′ ,n ′ ,j ′ ( m − m ′ ) h ψ ( j ) m,n , Y ψ ( j ′ ) m ′ ,n ′ i| ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ | . Since S λ is diagonal in the basis { ψ ( j ) m,n } it will significantly simplify our arguments if we replacethe decay provided by h X − λ i − / with S λ . Using Lemma E.1 we have that kh X − λ i − / [ P XP, P ˜ XP ] h X − λ i − / k≤ kh X − λ i − / P [ P XP, P ˜ XP ] P h X − λ i − / k ≤ kh X − λ i − / P S − λ kk S λ [ P XP, P ˜ X P ] S λ kk S − λ P h X − λ i − / k≤ C k S λ [ P XP, P ˜ XP ] S λ k Similar calculations for Y show that kh X − λ i − / [ P Y P, P ˜ XP ] h X − λ i − / k ≤ C k S λ [ P Y P, P ˜ XP ] S λ k Hence it suffices to show that S λ [ P XP, P ˜ X P ] S λ and S λ [ P Y P, P ˜ XP ] S λ are both bounded. Usingthe expressions for [ P XP, P ˜ XP ] and [ P Y P, P ˜ XP ] from Equations (59) and (60), we get S λ [ P XP, P ˜ XP ] S λ = X m,n,j X m ′ ,n ′ ,j ′ ( m − m ′ ) | λ − m | / | λ − m ′ | / h ψ ( j ) m,n , Xψ ( j ′ ) m ′ ,n ′ i| ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ | S λ [ P Y P, P ˜ XP ] S λ = X m,n,j X m ′ ,n ′ ,j ′ ( m − m ′ ) | λ − m | / | λ − m ′ | / h ψ ( j ) m,n , Y ψ ( j ′ ) m ′ ,n ′ i| ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ | Therefore, to finish the proof of Proposition 8.4, we will prove the following proposition:
Proposition F.1. If { ψ ( j ) m,n } is an s -localized basis with s > / , then there exists an absoluteconstant C such that for all λ ∈ G (recall that G is the set of gaps defined in Equation (23) ) wehave (cid:13)(cid:13) X m,n,j X m ′ ,n ′ ,j ′ ( m − m ′ ) | λ − m | / | λ − m ′ | / h ψ ( j ) m,n , Xψ ( j ′ ) m ′ ,n ′ i| ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ | (cid:13)(cid:13) ≤ C (61) (cid:13)(cid:13) X m,n,j X m ′ ,n ′ ,j ′ ( m − m ′ ) | λ − m | / | λ − m ′ | / h ψ ( j ) m,n , Y ψ ( j ′ ) m ′ ,n ′ i| ψ ( j ) m,n ih ψ ( j ′ ) m ′ ,n ′ | (cid:13)(cid:13) ≤ C. (62)F.4. Proof of Proposition F.1.
In this section, we will only prove Equation (61), Equation (62)follows by analogous steps.To bound Equation (61), we will use Schur’s test. Recall that Schur’s test tells us that if T is alinear operator defined by the discrete kernel K ( m, n, j, m ′ , n ′ , j ′ ): T f ( m, n, j ) = X m ′ ,n ′ ,j ′ K ( m, n, j, m ′ , n ′ , j ′ ) f ( m ′ , n ′ , j ′ )and if for some real, positive functions p, q we have X m ′ ,n ′ ,j ′ | K ( m, n, j, m ′ , n ′ , j ′ ) | q ( m ′ , n ′ , j ′ ) ≤ αp ( m, n, j ) and X m,n p ( m, n, j ) | K ( m, n, j, m ′ , n ′ , j ′ ) | ≤ βq ( m ′ , n ′ , j ′ ) . then k T k ≤ √ αβ . Equation (61) can be viewed as the operator norm of an operator defined by thefollowing discrete kernel: K ( m, n, j, m ′ , n ′ , j ′ ) := ( m − m ′ ) | λ − m | / | λ − m ′ | / h ψ ( j ) m,n , Xψ ( j ′ ) m ′ ,n ′ i . Choosing p ( m, n, j ) = | λ − m | − / and q ( m ′ , n ′ , j ′ ) = | λ − m ′ | − / and applying Schur’s test we seethat it is enough to find α, β such that X m ′ ,n ′ ,j ′ | m ′ − m || λ − m ′ || λ − m | / |h ψ ( j ) m,n , Xψ ( j ′ ) m ′ ,n ′ i| ≤ α | λ − m | / X m,n,j | m ′ − m || λ − m ′ | / | λ − m | |h ψ ( j ) m,n , Xψ ( j ′ ) m ′ ,n ′ i| ≤ β | λ − m ′ | / Multiplying both sides of the first inequality by | λ − m | / gives that we need to show that:(63) X m ′ ,n ′ ,j ′ | m ′ − m || λ − m ′ | |h ψ ( j ) m,n , Xψ ( j ′ ) m ′ ,n ′ i| ≤ α Similarly, multiplying both sides of the second inequality by | λ − m ′ | / gives that we need to showthat:(64) X m,n,j | m ′ − m || λ − m | |h ψ ( j ) m,n , Xψ ( j ′ ) m ′ ,n ′ i| ≤ β Since X is self-adjoint, proving the bound in Equation (63) immediately implies Equation (64) with α = β by performing the change of index ( m, n, j ) ↔ ( m ′ , n ′ , j ′ ). Hence, we will focus on Equation(63) for the remainder of this section.Similar to the proof of Proposition 6.2 in Appendix D, our main technique for proving Equation(63) will be inserting a partition unity of the form: X k,ℓ χ k,ℓ ( x ) = 1where χ k,ℓ ( x ) = ( x ∈ (cid:2) k − , k + (cid:1) × (cid:2) ℓ − , ℓ + (cid:1) . Since { ψ ( j ) m,n } is an orthonormal basis h ψ ( j ) m,n , ψ ( j ′ ) m ′ ,n ′ i = 0 whenever ( m, n, j ) = ( m ′ , n ′ , j ′ ). There-fore, we easily see that X m ′ ,n ′ ,j ′ | m ′ − m || λ − m ′ | |h ψ ( j ) m,n , Xψ ( j ′ ) m ′ ,n ′ i| = X m ′ ,n ′ ,j ′ | m ′ − m || λ − m ′ | |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i| . Now we can insert our partition of unity to get X m ′ ,n ′ ,j ′ | m ′ − m || λ − m ′ | |h ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i|≤ X m ′ ,n ′ ,j ′ X k,ℓ | m ′ − m || λ − m ′ | |h χ k,ℓ ψ ( j ) m,n , ( X − m ) ψ ( j ′ ) m ′ ,n ′ i|≤ X m ′ ,n ′ ,j ′ X k,ℓ | m ′ − m || λ − m ′ | k χ k,ℓ ( X − m ) ψ ( j ) m,n kk χ k,ℓ ψ ( j ′ ) m ′ ,n ′ k≤ M sup j ′ X m ′ ,n ′ X k,ℓ | m ′ − m || λ − m ′ | k χ k,ℓ ( X − m ) ψ ( j ) m,n kk χ k,ℓ ψ ( j ′ ) m ′ ,n ′ k (65)Using the fact that | m − m ′ | ≤ | m − k | + | m ′ − k | , we can now upper bound Equation (65) by thesum of the following two terms: X m ′ ,n ′ X k,ℓ | m − k || λ − m ′ | k χ k,ℓ ( X − m ) ψ ( j ) m,n kk χ k,ℓ ψ ( j ′ ) m ′ ,n ′ k (66) X m ′ ,n ′ X k,ℓ | m ′ − k || λ − m ′ | k χ k,ℓ ( X − m ) ψ ( j ) m,n kk χ k,ℓ ψ ( j ′ ) m ′ ,n ′ k . (67)We will bound Equation (66) in Appendix F.4.1 and Equation (67) in Appendix F.4.2.F.4.1. Bounding Equation (66) . For this proof, we will fix a choice of ( m, n, j ) and prove a boundwhich is uniform in ( m, n, j ). Using Lemma C.1 with s = 1 / ǫ and s = 1 + ǫ we have that forany ǫ > k χ k,ℓ ψ ( j ′ ) m ′ ,n ′ k ≤ C k χ k,ℓ ( | X − m ′ | + 1) / ǫ ( | Y − n ′ | + 1) ǫ ψ ( j ′ ) m ′ ,n ′ kh m ′ − k i / ǫ h n ′ − ℓ i ǫ Next, using Lemma C.2 k χ k,ℓ ( | X − m ′ | + 1) / ǫ ( | Y − n ′ | + 1) ǫ ψ ( j ′ ) m ′ ,n ′ k≤ k ( | X − m ′ | + 1) / ǫ ψ ( j ′ ) m ′ ,n ′ k + k ( | Y − n ′ | + 1) / ǫ ψ ( j ′ ) m ′ ,n ′ k )but this quantity is bounded by a constant, C , since we assume that the basis { ψ ( j ) m,n } is s -localizedwith s > /
2. Therefore, we can upper bound Equation (66) with X m ′ ,n ′ X k,ℓ | m − k || λ − m ′ | k χ k,ℓ ( X − m ) ψ ( j ) m,n kk χ k,ℓ ψ ( j ′ ) m ′ ,n ′ k≤ C X m ′ ,n ′ X k,ℓ | m − k || λ − m ′ | k χ k,ℓ ( X − m ) ψ ( j ) m,n kh m ′ − k i / ǫ h n ′ − ℓ i ǫ Applying Lemma C.1 with s = 1 and s = 1 / ǫ we also have that k χ k,ℓ ( X − m ) ψ ( j ) m,n k≤ C k χ k,ℓ ( | X − m | + 1) ( | Y − n | + 1) / ǫ ψ ( j ) m,n kh m − k ih n − ℓ i / ǫ (69)To reduce clutter, in the next few steps let us define:(70) A k,ℓ := k χ k,ℓ ( | X − m | + 1) ( | Y − n | + 1) / ǫ ψ ( j ) m,n k . Note that we have excluded the dependence on ( m, n, j ) in our notation since we have fixed a choice( m, n, j ) for this proof. With this definition and the bound from Equation (69) we have that X m ′ ,n ′ X k,ℓ | m − k || λ − m ′ | k χ k,ℓ ( X − m ) ψ ( j ) m,n kh m ′ − k i / ǫ h n ′ − ℓ i ǫ ≤ C C X m ′ ,n ′ X k,ℓ | m − k || λ − m ′ | A k,ℓ h m − k ih n − ℓ i / ǫ h m ′ − k i / ǫ h n ′ − ℓ i ǫ ≤ C C X m ′ ,n ′ X k,ℓ | λ − m ′ | A k,ℓ h n − ℓ i / ǫ h m ′ − k i / ǫ h n ′ − ℓ i ǫ ≤ C C X n ′ ,ℓ h n − ℓ i / ǫ h n ′ − ℓ i ǫ X m ′ ,k A k,ℓ | λ − m ′ |h m ′ − k i / ǫ (71)Let’s focus our attention on the sum over ( m ′ , k ) X m ′ ,k A k,ℓ | λ − m ′ |h m ′ − k i / ǫ LGEBRAIC LOCALIZATION IMPLIES EXPONENTIAL LOCALIZATION IN NON-PERIODIC INSULATORS 53
Since λ , ℓ , and ( m, n, j ) are fixed this is a sum of the form X m ′ X k a [ k ] b [ m ′ ] c [ m ′ − k ]which is clearly the ℓ -norm of a convolution. Therefore, by Young’s convolution inequality (The-orem 3) with p = 2, q = 1 + ǫ and r = (2 − p − q ) − = ǫ )2+3 ǫ we have that X m ′ ,k A k,ℓ | λ − m ′ |h m ′ − k i / ǫ ≤ X k A k,ℓ ! / X m ′ | λ − m ′ | ǫ/ ! /q X m ′ h m ′ i (1 / ǫ ) r ! /r (72)It’s easy to check that (cid:18)
12 + ǫ (cid:19) r = 2 + 5 ǫ + 2 ǫ ǫ = 1 + ǫ + O ( ǫ )so for ǫ > C .Therefore, we conclude that(73) X m ′ ,k A k,ℓ | λ − m ′ |h m ′ − k i / ǫ ≤ C X k A k,ℓ ! / . Using this bound in Equation (71) then gives: X m ′ ,n ′ X k,ℓ | m − k || λ − m ′ | k χ k,ℓ ( X − m ) ψ ( j ) m,n kh m ′ − k i / ǫ h n ′ − ℓ i ǫ ≤ C C C X n ′ ,ℓ h n − ℓ i / ǫ h n ′ − ℓ i ǫ X k A k,ℓ ! / . Similar to before, this sum is the ℓ -norm of a convolution in n ′ , ℓ . Therefore, by Young’s convolutioninequality (Theorem 3) with p = q = 2, r = 1 we have that X n ′ ,ℓ h n − ℓ i / ǫ h n ′ − ℓ i ǫ X k A k,ℓ ! / ≤ X k,ℓ A k,ℓ / X ℓ h n − ℓ i ǫ ! / X n ′ h n ′ i ǫ ! The last two sums are clearly bounded for any ǫ > A k,ℓ as Equation (70): X k,ℓ A k,ℓ = X k,ℓ k χ k,ℓ ( | X − m | + 1) ( | Y − n | + 1) / ǫ ψ ( j ) m,n k = k ( | X − m | + 1) ( | Y − n | + 1) / ǫ ψ ( j ) m,n k . Applying Lemma C.2, we see that this quantity is bounded by a constant so long as we assumethat { ψ ( j ) m,n } is bounded with s > /
2. This finishes the proof that Equation (66) is bounded.
F.4.2.
Bounding Equation (67) . This bound follows by essentially the same argument as used tobound Equation (66); we only include it for completeness. Similar to before, we will fix a choice of( m, n, j ) and prove a bound which is uniform in ( m, n, j ).Using Lemma C.1 with s = 3 / ǫ and s = 1 / ǫ we have that for any ǫ > k χ k,ℓ ψ ( j ′ ) m ′ ,n ′ k ≤ C k χ k,ℓ ( | X − m ′ | + 1) / ǫ ( | Y − n ′ | + 1) / ǫ ψ ( j ′ ) m ′ ,n ′ kh m ′ − k i / ǫ h n ′ − ℓ i / ǫ Next, using Lemma C.2 k χ k,ℓ ( | X − m ′ | + 1) / ǫ ( | Y − n ′ | + 1) / ǫ ψ ( j ′ ) m ′ ,n ′ k≤ k ( | X − m ′ | + 1) ǫ ψ ( j ′ ) m ′ ,n ′ k + k ( | Y − n ′ | + 1) ǫ ψ ( j ′ ) m ′ ,n ′ k but this quantity is bounded by a constant, C , since we assume that the basis { ψ ( j ) m,n } is s -localizedwith s > /
2. Therefore, we can upper bound Equation (67) with X m ′ ,n ′ X k,ℓ | m ′ − k || λ − m ′ | k χ k,ℓ ( X − m ) ψ ( j ) m,n kk χ k,ℓ ψ ( j ′ ) m ′ ,n ′ k≤ C X m ′ ,n ′ X k,ℓ | m ′ − k || λ − m ′ | k χ k,ℓ ( X − m ) ψ ( j ) m,n kh m ′ − k i / ǫ h n ′ − ℓ i / ǫ ≤ C X m ′ ,n ′ X k,ℓ | λ − m ′ | k χ k,ℓ ( X − m ) ψ ( j ) m,n kh m ′ − k i / ǫ h n ′ − ℓ i / ǫ Applying Lemma C.1 with s = 0 and s = 1 + ǫ we also have that k χ k,ℓ ( X − m ) ψ ( j ) m,n k≤ C k χ k,ℓ ( | X − m | + 1)( | Y − n | + 1) ǫ ψ ( j ) m,n kh n − ℓ i ǫ (75)To reduce clutter, in the next few steps let us define (recall that we have fixed ( m, n, j )):(76) ˜ A k,ℓ := k χ k,ℓ ( | X − m | + 1)( | Y − n | + 1) ǫ ψ ( j ) m,n k . With this definition and the bound from Equation (75), we have that X m ′ ,n ′ X k,ℓ | λ − m ′ | k χ k,ℓ ( X − m ) ψ ( j ) m,n kh m ′ − k i / ǫ h n ′ − ℓ i / ǫ ≤ C C X m ′ ,n ′ X k,ℓ | λ − m ′ | ˜ A k,ℓ h n − ℓ i ǫ h m ′ − k i / ǫ h n ′ − ℓ i / ǫ ≤ C C X n ′ ,ℓ h n − ℓ i ǫ h n ′ − ℓ i / ǫ (cid:18)X m ′ ,k ˜ A k,ℓ | λ − m ′ |h m ′ − k i / ǫ (cid:19) (77)Using our calculation from Appendix F.4.1 (Equation (73)), we know that X m ′ ,n ′ X k,ℓ | λ − m ′ | k χ k,ℓ ( X − m ) ψ ( j ) m,n kh m ′ − k i / ǫ h n ′ − ℓ i / ǫ ≤ C X n ′ ,ℓ h n − ℓ i ǫ h n ′ − ℓ i / ǫ (cid:16)X k ˜ A k,ℓ (cid:17) / ≤ (cid:16)X k,ℓ ˜ A k,ℓ (cid:17) / X ℓ h n − ℓ i ǫ ! X n ′ h n ′ i ǫ ! / , where in the last line we have used Young’s convolution inequality (Theorem 3) with p = q = 2and r = 1. Recalling the definition of ˜ A k,ℓ we have that X k,ℓ ˜ A k,ℓ = X k,ℓ k χ k,ℓ ( | X − m | + 1)( | Y − n | + 1) ǫ ψ ( j ) m,n k = k ( | X − m | + 1)( | Y − n | + 1) ǫ ψ ( j ) m,n k which is bounded by a constant due to Lemma C.2 and the assumption that { ψ ( j ) m,n } is s -localizedwith s > /
2. This finishes the proof that Equation (67) is bounded, completing the proof ofProposition F.1.
References [1] Christian Brouder, Gianluca Panati, Matteo Calandra, Christophe Mourougane, and Nicola Marzari. Exponentiallocalization of Wannier functions in insulators.
Phys. Rev. Lett. , 98(4):046402, 2007.[2] Jacques Des Cloizeaux. Energy bands and projection operators in a crystal: Analytic and asymptotic properties.
Phys. Rev. , 135:A685–A697, Aug 1964.[3] Horia D Cornean, Domenico Monaco, and Massimo Moscolari. Beyond diophantine wannier diagrams: gaplabelling for bloch-landau hamiltonians. arXiv preprint arXiv:1810.05623 , 2018.[4] Jacques Des Cloizeaux. Analytical properties of n-dimensional energy bands and Wannier functions.
Phys. Rev. ,135(3A):A698, 1964.[5] Matthew B Hastings. Making almost commuting matrices commute.
Commun. Math. Phys. , 291(2):321–345,2009.[6] Bernard Helffer and Johannes Sj¨ostrand.
Analyse semi-classique pour l’´equation de Harper:(avec application `al’´equation de Schr¨odinger avec champ magn´etique) . Soci´et´e math´ematique de France, 1988.[7] Steven Kivelson. Wannier functions in one-dimensional disordered systems: Application to fractionally chargedsolitons.
Phys. Rev. B: Condens. Matter , 26(8):4269, 1982.[8] Walter Kohn. Analytic properties of Bloch waves and Wannier functions.
Phys. Rev. , 115(4):809, 1959.[9] E.H. Lieb and M. Loss.
Analysis . American Mathematical Society, 2001.[10] Giovanna Marcelli, Domenico Monaco, Massimo Moscolari, and Gianluca Panati. The Haldane model and itslocalization dichotomy. arXiv preprint arXiv:1909.03298 , 2019.[11] Giovanna Marcelli, Massimo Moscolari, and Gianluca Panati. Localization implies Chern triviality in non-periodic insulators. arXiv preprint arXiv:2012.14407 , 2020.[12] Nicola Marzari, Arash A Mostofi, Jonathan R Yates, Ivo Souza, and David Vanderbilt. Maximally localizedWannier functions: Theory and applications.
Rev. Mod. Phys. , 84(4):1419, 2012.[13] Domenico Monaco, Gianluca Panati, Adriano Pisante, and Stefan Teufel. Optimal decay of Wannier functionsin Chern and quantum Hall insulators.
Commun. Math. Phys. , 359(1):61–100, 2018.[14] A Nenciu and Gheorghe Nenciu. The existence of generalised Wannier functions for one-dimensional systems.
Commun. Math. Phys. , 190(3):541–548, 1998.[15] Gheorghe Nenciu. Existence of the exponentially localised Wannier functions.
Commun. Math. Phys. , 91(1):81–85, 1983.[16] Gheorghe Nenciu. Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effectivehamiltonians.
Rev. Mod. Phys. , 63:91–127, Jan 1991.[17] Qian Niu. Theory of the quantized adiabatic particle transport.
Mod. Phys. Lett. B , 5:923–931, 1991.[18] Gianluca Panati. Triviality of bloch and bloch–dirac bundles. In
Ann. Inst. Henri Poincare , volume 8, pages995–1011. Springer, 2007.[19] Michael Reed and Barry Simon.
II: Fourier Analysis, Self-Adjointness , volume 2. Elsevier, 1975.[20] Kevin D Stubbs, Alexander B Watson, and Jianfeng Lu. Existence and computation of generalized Wannierfunctions for non-periodic systems in two dimensions and higher. arXiv preprint arXiv:2003.06676 , 2020.[21] Kevin D Stubbs, Alexander B Watson, and Jianfeng Lu. The iterated projected position algorithm for con-structing exponentially localized generalized Wannier functions for periodic and non-periodic insulators in twodimensions and higher. arXiv preprint arXiv:2010.01434 , 2020. (JL) Department of Mathematics, Department of Physics, and Department of Chemistry, DukeUniversity, Box 90320, Durham, NC 27708, USA
Email address : [email protected] (KDS) Department of Mathematics, Duke University, Box 90320, Durham, NC 27708, USA Email address ::