Eigensystem multiscale analysis for Anderson localization in energy intervals
aa r X i v : . [ m a t h - ph ] N ov EIGENSYSTEM MULTISCALE ANALYSIS FOR ANDERSONLOCALIZATION IN ENERGY INTERVALS
ALEXANDER ELGART AND ABEL KLEIN
Abstract.
We present an eigensystem multiscale analysis for proving localiza-tion (pure point spectrum with exponentially decaying eigenfunctions, dynam-ical localization) for the Anderson model in an energy interval. In particular,it yields localization for the Anderson model in a nonempty interval at thebottom of the spectrum. This eigensystem multiscale analysis in an energyinterval treats all energies of the finite volume operator at the same time, es-tablishing level spacing and localization of eigenfunctions with eigenvalues inthe energy interval in a fixed box with high probability. In contrast to theusual strategy, we do not study finite volume Green’s functions. Instead, weperform a multiscale analysis based on finite volume eigensystems (eigenvaluesand eigenfunctions). In any given scale we only have decay for eigenfunctionswith eigenvalues in the energy interval, and no information about the othereigenfunctions. For this reason, going to a larger scale requires new argumentsthat were not necessary in our previous eigensystem multiscale analysis for theAnderson model at high disorder, where in a given scale we have decay for alleigenfunctions.
Contents
Introduction 21. Main results 52. Localization at the bottom of the spectrum 102.1. Fixed disorder 102.2. Fixed interval 113. Preamble to the eigensystem multiscale analysis 113.1. Subsets, boundaries, etc. 113.2. Lemmas for energy intervals 123.3. Lemmas for the multiscale analysis 143.4. Suitable covers of a box 233.5. Probability estimate for level spacing 244. Eigensystem multiscale analysis 245. Localization 316. Connection with the Green’s functions multiscale analysis 34References 37
Date : Version of September 19, 2018.A.E. was supported in part by the NSF under grant DMS-1210982.A.K. was supported in part by the NSF under grant DMS-1001509.
Introduction
We present an eigensystem multiscale analysis for proving localization (purepoint spectrum with exponentially decaying eigenfunctions, dynamical localization)for the Anderson model in an energy interval. In particular, it yields localizationfor the Anderson model in a nonempty interval at the bottom of the spectrum.The well known methods developed for proving localization for random Schr¨odingeroperators, the multiscale analysis [FroS, FroMSS, Dr, DrK, S, CoH, FK2, GK1, Kl,BoK, GK4] and the fractional moment method [AM, A, ASFH, AENSS, AiW], arebased on the study of finite volume Green’s functions. Multiscale analyses basedon Green’s functions are performed either at a fixed energy in a single box, or forall energies but with two boxes with an ‘either or’ statement for each energy.In [EK] we provided an implementation of a multiscale analysis for the Ander-son model at high disorder based on finite volume eigensystems (eigenvalues andeigenfunctions). In contrast to the usual strategy, we did not study finite volumeGreen’s functions. Information about eigensystems at a given scale was used to de-rive information about eigensystems at larger scales. This eigensystem multiscaleanalysis treats all energies of the finite volume operator at the same time, giving acomplete picture in a fixed box. For this reason it does not use a Wegner estimateas in a Green’s functions multiscale analysis, it uses instead a probability estimatefor level spacing derived by Klein and Molchanov from Minami’s estimate [KlM,Lemma 2]. This eigensystem multiscale analysis for the Anderson model at highdisorder has been enhanced in [KlT] by a bootstrap argument as in [GK1, Kl].The motivation for developing an alternative approach to localization is relatedto a new focus among the mathematical physics community in disordered systemswith an infinite number of particles, for which Green’s function methods breakdown. The direct study of the structure of eigenfunctions for such systems has beenadvocated by Imbrie [I1, I2] in a context of both single and many-body localization.The Green’s function methods allow for proving localization in energy intervals,and hence localization has also been proved at fixed disorder in an interval at theedge of the spectrum (or, more generally, in the vicinity of a spectral gap), andfor a fixed interval of energies at the bottom of the spectrum for sufficiently highdisorder. (See, for example, [HM, KSS, FK1, ASFH, GK2, K, GK4, AiW].) Thesemethods do not differentiate between energy intervals and the whole spectrum; theycan be used whenever the initial step can be established.The results in [EK] yield localization for the Anderson model in the whole spec-trum, which in practice requires high disorder. This eigensystem multiscale analysistreats all energies of the finite volume operator at the same time, at a given scale wehave decay for all eigenfunctions, and the induction step uses information about alleigenvalues and eigenfunctions. The method does not have a straightforward exten-sion for proving localization in an energy interval, since at any give scale we wouldonly have information (decay) about eigenfunctions corresponding to eigenvaluesin the given interval. For this reason, when performing an eigenfunction multiscaleanalysis in an energy interval, going to a larger scale requires new arguments thatwere not necessary in our previous eigensystem multiscale analysis for the Andersonmodel at high disorder, where in a given scale we have decay for all eigenfunctions.In this paper we develop a version of the eigensystem multiscale analysis tailoredto the establishment of localization for the Anderson model in an energy interval.This version yields localization at fixed disorder on an interval at the edge of the
IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 3 spectrum (or in the vicinity of a spectral gap), and at a fixed interval at the bottomof the spectrum for sufficiently high disorder.The Anderson model is a random Schr¨odinger operator H ω on ℓ ( Z d ) (see Def-inition 1.5). Multiscale analyses prove statements about finite volume operators H ω , Λ , the restrictions of H ω to finite boxes Λ. The eigensystem multiscale analysisdeveloped in this article establishes eigensystem localization in a bounded energyinterval with good probability at large scales, as we will now explain.An eigensystem { ( ϕ j , λ j ) } j ∈ J for H ω , Λ consists of eigenpairs ( ϕ j , λ j ), where λ j is an eigenvalue for H ω , Λ and ϕ j is a corresponding normalized eigenfunction,such that { ϕ j } j ∈ J is an orthonormal basis for the finite dimensional Hilbert space ℓ (Λ). If all eigenvalues of H ω , Λ are simple, we can rewrite the eigensystem as { ( ϕ λ , λ ) } λ ∈ σ ( H ω , Λ ) .We define eigensystem localization in a bounded energy interval I in the followingway. We fix appropriate exponents β, τ ∈ (0 ,
1) (see (1.1)), take m >
0, and saythat a box Λ of side L is ( m, I )-localizing for H ω (see Definition 1.3) if Λ is levelspacing (i.e., the eigenvalues of H ω , Λ are simple and separated by at least e − L β ), andeigenfunctions corresponding to eigenvalues in the interval I decay exponentially asfollows: if λ ∈ σ ( H ω , Λ ) ∩ I , then there exists x λ ∈ Λ such that the correspondingeigenfunction ϕ λ satisfies | ϕ λ ( y ) | ≤ e − mh I ( λ ) k y − x λ k for all y ∈ Λ with k y − x λ k ≥ L τ , where h I (defined in (1.12)) is a concave function on I , taking the value one at thecenter of the interval and the value zero at the endpoints. The modulation of thedecay of the eigenfunctions by the function h I is a new feature of our method.Our multiscale analysis shows that eigenfunction localization in an energy inter-val with good probability at some large enough scale implies eigenfunction local-ization with good (scale dependent and improving as the scale grows) probabilityfor all sufficiently large scales, in a slightly smaller energy interval. The key stepshows that localization at a large scale ℓ yields localization at a much larger scale L . The proof proceeds by covering a box Λ L of side L by boxes of side ℓ , whichare mostly ( m, I )-localizing, and showing this implies that Λ L is ( m ′ , I ′ )-localizing.There are always some losses, m ′ < m and I ′ ( I , but this losses are controllable,and continuing this procedure we converge to some rate of decay m ∞ > I ∞ = ∅ .The eigensystem multiscale analysis in an energy interval I requires a new ingre-dient, absent in the treatment of the system at high disorder given in [EK], where I = R and h I = 1. In broad terms, the reason is that our energy interval multi-scale scheme only carries information about eigenfunctions with eigenvalues in theinterval I , and contains no information whatsoever concerning eigenfunctions witheigenvalues that lie outside the interval I . Given boxes Λ ℓ ⊂ Λ L , with ℓ ≪ L , acrucial step in our analysis shows that if ( ψ, λ ) is an eigenpair for H ω , Λ L , with λ ∈ I not too close to the eigenvalues of H ω , Λ ℓ corresponding to eigenfunctions localizeddeep inside Λ ℓ , and the box Λ ℓ is ( m, I )-localizing for H ω , then ψ is exponentiallysmall deep inside Λ ℓ (see Lemma 3.4(ii)). This is proven by expanding the values of ψ in Λ ℓ in terms of the ( m, I )-localizing eigensystem { ( ϕ ν , ν ) } ν ∈ σ ( H ω , Λ ℓ ) for H ω , Λ ℓ .The difficulty is that we only have decay for the eigenfunctions ϕ ν with ν ∈ I ; weknow nothing about ϕ ν if ν / ∈ I . We overcame this difficulty by showing that thedecay of the term containing the latter eigenfunctions comes from the distance from ALEXANDER ELGART AND ABEL KLEIN the eigenvalue λ to the complement of the interval I , using Lemmas 3.2 and 3.3.As a result, it is natural to expect that the decay rate for the localization of eigen-functions goes to zero as the eigenvalues approach the edges of the interval I . Theintroduction of the modulating function h I in the decay models this phenomenon.The same difficulty appears if, given an ( m, I )-localizing box Λ for H ω , we tryto recover the decay of the Green’s function at an energy λ ∈ I not too close to theeigenvalues of H ω , Λ . The simplest approach is to decompose the Green’s functionin terms of an ( m, I )-localizing eigensystem { ( ϕ ν , ν ) } ν ∈ σ ( H ω , Λ ) for H ω , Λ : (cid:10) δ x , ( H ω , Λ − λ ) − δ y (cid:11) = X ν ∈ σ ( H ω , Λ ) ( ν − λ ) − ϕ ν ( x ) ϕ ν ( y ) . The sum over the eigenvalues inside the interval I can be estimated using the decayof the corresponding eigenfunctions, but we have a problem estimating the sum overeigenvalues outside I since we have no information concerning the spatial decayproperties of the corresponding eigenfunctions. To overcome this difficulty, we usea more delicate argument (see Lemma 6.4) that decomposes the Green’s functioninto a sum of two analytic functions of H ω , Λ with appropriate decay properties (seeLemmas 3.2 and 3.3 for details), obtaining the desired decay of the Green function: (cid:12)(cid:12)(cid:10) δ x , ( H ω , Λ − λ ) − δ y (cid:11)(cid:12)(cid:12) ≤ e − m ′ h I ( λ ) k x − y k . Readers familiar with the Green’s function multiscale analysis may notice thatthe modulation by the function h I is not required there. This has to do with the factthe Green’s function approach essentially considers each energy value separately,while the eigensystem approach treats the whole energy interval simultaneously. AGreen’s function multiscale analysis is performed at a fixed energy; the modulationof the decay may appear in the starting condition, but not in the multiscale analysisproper. (The starting condition near an spectral edge is usually obtained from theCombes-Thomas estimate, which modulates the decay rate by the distance to thespectral edge.)A version of our main result, Theorem 1.6, can be stated as follows. (Theexponents ζ, ξ ∈ (0 ,
1) and γ > L ( x ) denotes the box in Z d ofside L centered at x ∈ R d as in (1.10).) Theorem (Eigensystem multiscale analysis) . Let H ω be an Anderson model. Let I = ( E − A , E + A ) ⊂ R , with E ∈ R and A > , and < m ≤ log (cid:0) A d (cid:1) .Suppose for some scale L we have inf x ∈ R d P { Λ L ( x ) is ( m , I ) -localizing for H ω } ≥ − e − L ζ . Then, if L is sufficiently large, there exist m ∞ = m ∞ ( L ) > and A ∞ = A ∞ ( L ) ∈ (0 , A ) , with lim L →∞ A ∞ ( L ) = A and lim L →∞ m ∞ ( L ) = m ,such that, setting I ∞ = ( E − A ∞ , E + A ∞ ) , we have inf x ∈ R d P { Λ L ( x ) is ( m ∞ , I ∞ ) -localizing for H ω } ≥ − e − L ξ , for all L ≥ L γ . The theorem yields all the usual forms of Anderson localization on the interval I ∞ . In particular we obtain the following version of Corollary 1.8. Corollary (Localization in an energy interval) . Suppose the theorem holds for anAnderson model H ω . Then the following holds with probability one: IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 5 (i) H ω has pure point spectrum in the interval I ∞ . (ii) If ψ λ is a normalized eigenfunction of H ω with eigenvalue λ ∈ I ∞ , then ψ λ is exponentially localized with rate of decay m ∞ h I ∞ ( λ ) , more precisely, | ψ λ ( x ) | ≤ C ω ,λ e − m ∞ h I ∞ ( λ ) k x k for all x ∈ Z d . In particular, our results prove localization at the bottom of the spectrum. Let H ω be an Anderson model, and let E be the bottom of the almost sure spectrum of H ω . We consider intervals at the bottom of the spectrum, more precisely, intervalsof the form J = [ E , E + A ) with A >
0. We set ˜ J = ( E − A, E + A ), so J ∩ Σ = ˜ J ∩ Σ, call a box ( m, J )-localizing if it is ( m, ˜ J )-localizing, etc. Thefollowing is a version of Theorem 2.2. Theorem (Localization at the bottom of the spectrum) . Let H ω be an Andersonmodel, and fix < ξ < ζ < dd +2 . Then there is γ > , such that, if L is sufficientlylarge, there exist m ζ, ∞ = m ζ, ∞ ( L ) > and A ζ, ∞ = A ζ, ∞ ( L ) ∈ (0 , A ) , with lim L →∞ A ζ, ∞ ( L ) L ζd = C d,µ and lim L →∞ m ∞ ( L ) L ζd = 19 d C d,µ , such that, setting J ζ, ∞ = [ E , E + A ζ, ∞ ) , for all L ≥ L γ we have inf x ∈ R d P { Λ L ( x ) is ( m ζ, ∞ , J ζ, ∞ ) -localizing for H ω } ≥ − e − L ξ , In particular, the conclusions of the Corollary hold in the interval J ζ, ∞ . We also establish localization in a fixed interval at the bottom of the the spec-trum, for sufficiently large disorder (Theorem 2.3).Our main results and definitions are stated in Section 1. Theorem 1.6 is ourmain result, which we prove in Section 4. Theorem 1.7, derived from Theorem 1.6,encapsulates localization in an energy interval for the Anderson model and yieldsCorollary 1.8, which contains typical statements of Anderson localization and dy-namical localization in an energy interval. Theorem 1.7 and Corollary 1.8 are provenin Section 5. In Section 2 we show how to fulfill the starting condition for Theo-rem 1.6 and establish localization in an interval at the bottom of the spectrum, forfixed disorder (Theorem 2.2) and in a fixed interval for sufficiently large disorder(Theorem 2.3). Section 3 contains notations, definitions and lemmas required forthe proof of the eigensystem multiscale analysis given in Section 4. The connectionwith the Green’s functions multiscale analysis is established in Section 6.1.
Main results
In this article we will use many positive exponents, which will be required tosatisfy certain relations. We consider ξ, ζ, β, τ ∈ (0 ,
1) and γ > < ξ < ζ < β < γ < < γ < q ζξ and max n γβ, ( γ − β +1 γ o < τ < , (1.1)and note that0 < ξ < ξγ < ζ < β < τγ < γ < τ < < − βτ − β < γ < τβ . (1.2)We also take e ζ = ζ + β ∈ ( ζ, β ) and e τ = 1 + τ ∈ ( τ, , (1.3) ALEXANDER ELGART AND ABEL KLEIN so ( γ − e ζ + 1 < ( γ − β + 1 < γτ. (1.4)We also consider κ ∈ (0 ,
1) and κ ′ ∈ [0 ,
1) such that κ + κ ′ < τ − γβ. (1.5)We set ̺ = min n κ, − τ , γτ − ( γ − e ζ − o , note 0 < κ ≤ ̺ < , (1.6)and choose ς ∈ (0 , − ̺ ] , so ̺ < − ς. (1.7)We consider these exponents fixed and do not make explicit the dependence ofconstants on them. We write χ A for the characteristic function of the set A . By aconstant we always mean a finite constant. We will use C a,b,... , C ′ a,b,... , C ( a, b, . . . ),etc., to denote a constant depending on the parameters a, b, . . . . Note that C a,b,... may denote different constants in different equations, and even in the same equation.Given a scale L ≥
1, we sets L = ℓ γ (i.e., ℓ = L γ ) , L τ = ⌊ L τ ⌋ , and L e τ = ⌊ L e τ ⌋ . If x = ( x , x , . . . , x d ) ∈ R d , we set | x | = | x | = (cid:16)P dj =1 x j (cid:17) , and k x k = | x | ∞ =max j =1 , ,...,d | x j | . If x ∈ R d and Ξ ⊂ R d , we set dist( x, Ξ) = inf y ∈ Ξ k y − x k . Thediameter of a set Ξ ⊂ R d is given by diam Ξ = sup x,y ∈ Ξ k y − x k . H we will always denote a discrete Schr¨odinger operator, that is, an operator H = − ∆ + V on ℓ ( Z d ), where where ∆ is the (centered) discrete Laplacian:(∆ ϕ )( x ) := X y ∈ Z d | y − x | =1 ϕ ( y ) for ϕ ∈ ℓ ( Z d ) , (1.8)and V is a bounded potential. Given Φ ⊂ Θ ⊂ Z d , we consider ℓ (Φ) ⊂ ℓ (Θ)by extending functions on Φ to functions on Θ that are identically 0 on Θ \ Φ. IfΘ ⊂ Z d and ϕ ∈ ℓ (Θ), we let k ϕ k = k ϕ k and k ϕ k ∞ = max y ∈ Θ | ϕ ( y ) | .Given Θ ⊂ Z d , we let H Θ be the restriction of χ Θ Hχ Θ to ℓ (Θ). We call( ϕ, λ ) an eigenpair for H Θ if ϕ ∈ ℓ (Θ) with k ϕ k = 1, λ ∈ R , and H Θ ϕ = λϕ .(In other words, λ is an eigenvalue for H Θ and ϕ is a corresponding normalizedeigenfunction.) A collection { ( ϕ j , λ j ) } j ∈ J of eigenpairs for H Θ will be called aneigensystem for H Θ if { ϕ j } j ∈ J is an orthonormal basis for ℓ (Θ). If all eigenvaluesof H Θ are simple, we can rewrite the eigensystem as { ( ψ λ , λ ) } λ ∈ σ ( H Θ ) .Given Θ ⊂ Z d , a function ψ : Θ → C is called a generalized eigenfunction for H Θ with generalized eigenvalue λ ∈ R if ψ is not identically zero and h ( H Θ − λ ) ϕ, ψ i = 0 for all ϕ ∈ ℓ (Θ) with finite support . (1.9)In this case we call ( ψ, λ ) a generalized eigenpair for H Θ . (Eigenfunctions aregeneralized eigenfunctions, but we do not require generalized eigenfunctions to bein ℓ (Θ).)For convenience we consider boxes in Z d centered at points of R d . The box in Z d of side L > x ∈ R d is given byΛ L ( x ) = Λ R L ( x ) ∩ Z d , where Λ R L ( x ) = (cid:8) y ∈ R d ; k y − x k ≤ L (cid:9) . (1.10) IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 7
By a box Λ L we will mean a box Λ L ( x ) for some x ∈ R d . It is easy to see that forall L ≥ x ∈ R d we have ( L − d < | Λ L ( x ) | ≤ ( L + 1) d . Definition 1.1.
Given
R > , a finite set Θ ⊂ Z d will be called R -level spacing for H if all eigenvalues of H Θ are simple and | λ − λ ′ | ≥ e − R β for all λ, λ ′ ∈ σ ( H Θ ) , λ = λ ′ .If Θ is a box Λ L and R = L , we will simply say that Λ L is level spacing for H . Definition 1.2.
Let Λ L be a box, x ∈ Λ L , and m ≥ . Then ϕ ∈ ℓ (Λ L ) is said tobe ( x, m ) -localized if k ϕ k = 1 and | ϕ ( y ) | ≤ e − m k y − x k for all y ∈ Λ L with k y − x k ≥ L τ . (1.11)Note that m = 0 is allowed in Definition 1.2. Definition 1.3.
Let J = ( E − B, E + B ) ⊂ I = ( E − A, E + A ) , where E ∈ R and < B ≤ A , be bounded open intervals with the same center, and let m > . A box Λ L will be called ( m, J, I ) -localizing for H if the following holds: (i) Λ L is level spacing for H . (ii) There exists an ( m, J, I ) -localized eigensystem for H Λ L , that is, an eigen-system { ( ϕ ν , ν ) } ν ∈ σ ( H Λ L ) for H Λ L such that for all ν ∈ σ ( H Λ L ) there is x ν ∈ Λ L such that ϕ ν is ( x ν , mχ J ( ν ) h I ( ν )) -localized, where the modulatingfunction h I is defined by h I ( t ) = h (cid:0) t − EA (cid:1) for t ∈ R , where h ( s ) = ( − s if s ∈ [0 , otherwise . (1.12) We will say that Λ L is ( m, I ) -localizing for H if Λ L is ( m, I, I ) -localizing for H . Note that h I ( t ) > ⇐⇒ t ∈ I , in particular h I = χ I h I . Since χ J h I ≥ h J , ifΛ L is ( m, J, I )-localizing for H it is also ( m, J )-localizing for H . Remark 1.4. In [EK] we had I = R and h R = 1 , and called a box Λ L m -localizingif it was level spacing for H and for all ν ∈ σ ( H Λ L ) there is x ν ∈ Λ L such that ϕ ν is ( x ν , m ) -localized. Given an interval I = ( E − A, E + A ) and scales ℓ, L >
1, we use the notation I ℓ = ( E − A (1 − ℓ − κ ) , E + A (1 − ℓ − κ )) , (1.13) I ℓ = ( E − A (1 − ℓ − κ ) − , E + A (1 − ℓ − κ ) − ) . We write I Lℓ = ( I ℓ ) L = (cid:0) I L (cid:1) ℓ , note that I ℓℓ = I , and observe that χ I ℓ h I ≥ ℓ − κ χ I ℓ , i.e., h I ( t ) ≥ − (1 − ℓ − κ ) ≥ ℓ − κ for all t ∈ I ℓ . (1.14) Definition 1.5.
The Anderson model is the random discrete Schr¨odinger operator H ω := − ∆ + V ω on ℓ ( Z d ) , (1.15) where V ω is a random potential: V ω ( x ) = ω x for x ∈ Z d , where ω = { ω x } x ∈ Z d isa family of independent identically distributed randoms variables, whose commonprobability distribution µ is non-degenerate with bounded support. We assume µ isH¨older continuous of order α ∈ ( , : S µ ( t ) ≤ Kt α for all t ∈ [0 , , (1.16) where K is a constant and S µ ( t ) := sup a ∈ R µ { [ a, a + t ] } is the concentration func-tion of the measure µ . ALEXANDER ELGART AND ABEL KLEIN
It follows from ergodicity (e.g., [K, Theorem 3.9]) that σ ( H ω ) = Σ := σ ( − ∆) + supp µ = [ − d, d ] + supp µ with probability one . (1.17)The eigensystem multiscale analysis in an energy interval yields the followingtheorem. Theorem 1.6.
Let H ω be an Anderson model. Given m − > , there exists a afinite scale L = L ( d, m − ) and a constant C d,m − > with the following property:Suppose for some scale L ≥ L we have inf x ∈ R d P { Λ L ( x ) is ( m , I ) -localizing for H ω } ≥ − e − L ζ , (1.18) where I = ( E − A , E + A ) ⊂ R , with E ∈ R and A > , and m − L − κ ′ ≤ m ≤ log (cid:0) A d (cid:1) . (1.19) Then for all L ≥ L γ we have inf x ∈ R d P (cid:26) Λ L ( x ) is ( m ∞ , I ∞ , I L γ ∞ ) -localizing for H ω (cid:27) ≥ − e − L ξ , (1.20) where, with ̺ as in (1.6) , A ∞ = A ∞ ( L ) = A ∞ Y k =0 (cid:16) − L − κγ k (cid:17) , I ∞ = ( E − A ∞ , E + A ∞ ) , (1.21) m ∞ = m ∞ ( L ) = m ∞ Y k =0 (cid:16) − C d,m − L − ̺γ k (cid:17) < log (cid:0) A ∞ d (cid:1) . In particular, lim L →∞ A ∞ ( L ) = A and lim L →∞ m ∞ ( L ) = m . Theorem 1.6 yields all the usual forms of localization on the interval I ∞ . Tostate these results, we fix ν > d , and for a ∈ Z d we let T a be the operator on ℓ ( Z d )given by multiplication by the function T a ( x ) := h x − a i ν , where h x i = q k x k .Since h a + b i ≤ √ h a ih b i , we have k T a T − b k ≤ ν h a − b i ν . A function ψ : Z d → C will be called a ν -generalized eigenfunction for the discrete Schr¨odinger operator H if ψ is a generalized eigenfunction and (cid:13)(cid:13) T − ψ (cid:13)(cid:13) < ∞ . ( (cid:13)(cid:13) T − ψ (cid:13)(cid:13) < ∞ if and onlyif (cid:13)(cid:13) T − a ψ (cid:13)(cid:13) < ∞ for all a ∈ Z d .) We let V ( λ ) denote the collection of ν -generalizedeigenfunctions for H with generalized eigenvalue λ ∈ R . Given λ ∈ R and a, b ∈ Z d ,we set W ( a ) λ ( b ) := ( sup ψ ∈V ( λ ) | ψ ( b ) |k T − a ψ k if V ( λ ) = ∅ . (1.22)It is easy to see that for all a, b, c ∈ Z d we have W ( a ) λ ( a ) ≤ , W ( a ) λ ( b ) ≤ h b − a i ν , and W ( a ) λ ( c ) ≤ ν h b − a i ν W ( b ) λ ( c ) . (1.23) Theorem 1.7.
Suppose the conclusions of Theorem 1.6 hold for an Anderson model H ω , and let I = I ∞ , m = m ∞ . There exists a finite scale L = L d,ν,m − such that,given L ≤ L ∈ N and a ∈ Z d , there exists an event Y L,a with the followingproperties: (i) Y L,a depends only on the random variables { ω x } x ∈ Λ L ( a ) , and P {Y L,a } ≥ − C e − L ξ . (1.24) IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 9 (ii) If ω ∈ Y L,a , for all λ ∈ I we have that ( L = ℓ γ ) max b ∈ Λ L ( a ) W ( a ) ω ,λ ( b ) > e − mh Iℓ ( λ ) L = ⇒ max y ∈ A L ( a ) W ( a ) ω ,λ ( y ) ≤ e − mh Iℓ ( λ ) k y − a k , (1.25) where A L ( a ) := (cid:8) y ∈ Z d ; L ≤ k y − a k ≤ L (cid:9) . (1.26) In particular, for all ω ∈ Y L,a and λ ∈ I we have W ( a ) ω ,λ ( a ) W ( a ) ω ,λ ( y ) ≤ e − mh Iℓ ( λ ) k y − a k for all y ∈ A L ( a ) . (1.27)Theorem 1.7 implies Anderson localization and dynamical localization, and more,as shown in [GK3, GK4, EK]. In particular, we get the following corollary. Corollary 1.8.
Suppose the conclusions of Theorem 1.6 hold for an Andersonmodel H ω , and let I = I ∞ , m = m ∞ . Then the following holds with probabilityone: (i) H ω has pure point spectrum in the interval I . (ii) If ψ λ is an eigenfunction of H ω with eigenvalue λ ∈ I , then ψ λ is expo-nentially localized with rate of decay mh I ( λ ) , more precisely, | ψ λ ( x ) | ≤ C ω ,λ (cid:13)(cid:13) T − ψ (cid:13)(cid:13) e − mh I ( λ ) k x k for all x ∈ R d . (1.28)(iii) If λ ∈ I , then for all x, y ∈ Z d we have W ( x ) ω ,λ ( x ) W ( x ) ω ,λ ( y ) ≤ C m, ω ,ν ( h I ( λ )) − ν e ( + ν ) mh I ( λ )(2 d log h x i ) ξ e − mh I ( λ ) k y − x k . (1.29)(iv) If λ ∈ I , then for ψ ∈ χ { λ } ( H ω ) and all x, y ∈ Z d we have | ψ ( x ) | | ψ ( y ) | (1.30) ≤ C m, ω ,ν ( h I ( λ )) − ν (cid:13)(cid:13) T − x ψ (cid:13)(cid:13) e ( + ν ) mh I ( λ )(2 d log h x i ) ξ e − mh I ( λ ) k y − x k ≤ ν C m, ω ,ν ( h I ( λ )) − ν (cid:13)(cid:13) T − ψ (cid:13)(cid:13) h x i ν e ( + ν ) mh I ( λ )(2 d log h x i ) ξ e − mh I ( λ ) k y − x k . (v) If λ ∈ I , then there exists x λ = x ω ,λ ∈ Z d , such that for ψ ∈ χ { λ } ( H ω ) and all x ∈ Z d we have | ψ ( x ) | ≤ C m, ω ,ν ( h I ( λ )) − ν (cid:13)(cid:13) T − x λ ψ (cid:13)(cid:13) e ( + ν ) mh I ( λ )(2 d log h x λ i ) ξ e − mh I ( λ ) k x − x λ k ≤ ν C m, ω ,ν ( h I ( λ )) − ν (cid:13)(cid:13) T − ψ (cid:13)(cid:13) h x λ i ν e ( + ν ) mh I ( λ )(2 d log h x λ i ) ξ e − mh I ( λ ) k x − x λ k . (1.31)In Corollary 1.8, (i) and (ii) are statements of Anderson localization, (iii) and (iv)are statements of dynamical localization ((iv) is called SUDEC (summable uniformdecay of eigenfunction correlations) in [GK3]), and (v) is SULE (semi-uniformlylocalized eigenfunctions; see [DJLS1, DJLS2]).We can also derive statements of localization in expectation, as in [GK3, GK4]. Localization at the bottom of the spectrum
We now discuss how to obtain the initial step for the eigensystem multiscaleanalysis at the bottom of the spectrum and prove localization. Let H ω be anAnderson model, and set E = inf Σ (see (1.17)), the bottom of the almost surespectrum of H ω . We will consider intervals at the bottom of the spectrum, moreprecisely, intervals of the form J = [ E , E + A ) with A >
0. We set ˜ J = ( E − A, E + A ), so J ∩ Σ = ˜ J ∩ Σ, call a box ( m, J )-localizing if it is ( m, ˜ J )-localizingas in Definition 1.3, etc. We also set J L and J L so f J L = ˜ J L and f J L = ˜ J L .2.1. Fixed disorder.Proposition 2.1.
Let H ω be an Anderson model, and set E = inf Σ . There existsa constant C d,µ > such that, given ζ ∈ (0 , , for sufficiently large L we have inf x ∈ R d P n H Λ L ( x ) > E + C d,µ L − ζd o ≥ − e − L ζ . (2.1) In particular, for all intervals J ζ ( L ) = [ E , E + C d,µ L − ζd ) and all m > we have inf x ∈ R d P { Λ L ( x ) is ( m, J ζ ( L )) -localizing for H ω } ≥ − e − L ζ . (2.2)The estimate (2.1) follows from a Lifshitz tails estimate. It can be derived from[K, Proof of Theorem 11.4]. Although the boxes in [K] are all centered at pointsin Z d , the arguments, including the crucial [K, Lemma 6.4], can be extended toboxes centered at points in R d . Note that (2.2) follows trivially from (2.1). Sincethe probability distribution µ is a continuous measure (see (1.16)), it follows from(1.17) that J ζ ( L ) ⊂ Σ for all sufficiently large L .We will now combine Proposition 2.1 with Theorem 1.6, taking I = ^ J ζ ( L ),i.e., E = E and A = C d,µ L − ζd in Theorem 1.6. To satisfy (1.19) for L large,we take m = d C d,µ L − ζd , m − = d C d,µ and κ ′ = ζd . To satisfy (1.5) we require ζd < τ − γβ , and then choose 0 < κ < τ − γβ − κ ′ . Since for a fixed ζ we can take τ and γ close to 1 and β close to ζ , respecting (1.1), we find we can choose theparameters in (1.1) as long as ζd < − ζ ⇐⇒ ζ < dd +2 . (2.3)We obtain the following theorem. Theorem 2.2.
Let H ω be an Anderson model, and fix < ξ < ζ < dd +2 . Thenthere exists γ > such that, if L is sufficiently large, for all L ≥ L γ we have inf x ∈ R d P (cid:26) Λ L ( x ) is ( m ζ, ∞ , J ζ, ∞ , J L γ ζ, ∞ ) -localizing for H ω (cid:27) ≥ − e − L ξ , (2.4) where A ζ, ∞ = A ζ, ∞ ( L ) = C d,µ L − ζd ∞ Y k =0 (cid:16) − L − κγ k (cid:17) ≥ C d,µ L − ζd , (2.5) J ζ, ∞ = [ E , E + A ζ, ∞ ) ⊃ [ E , E + C d,µ L − ζd ) ,m ζ, ∞ = m ζ, ∞ ( L ) = d C d,µ L − ζd ∞ Y k =0 (cid:16) − C d, d C d,µ L − ̺γ k (cid:17) ≥ d C d,µ L − ζd . IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 11
In particular, the conclusions of Theorem 1.7 and Corollary 1.8 hold in the interval J ζ, ∞ . Fixed interval.
We may also use disorder to start the eigensystem multiscaleanalysis in a fixed interval at the bottom of the the spectrum. To do so we introducea disorder parameter g >
0, and set H g, ω = − ∆ + gV ω . We assume { } ∈ supp µ ⊂ [0 , ∞ ), so it follows from (1.17) that E = − d . Then, given B > ζ ∈ (0 , x ∈ R d P (cid:8) H g, Λ L ( x ) ≥ − d + B (cid:9) ≥ inf x ∈ R d P { gω x ≥ B for all x ∈ Λ L ( x ) } (2.6) ≥ (cid:0) − µ ([0 , g − B )) (cid:1) ( L +1) d ≥ (cid:0) − K ( g − B ) α (cid:1) ( L +1) d ≥ − ( L + 1) d K ( g − B ) α ≥ − L − ζ for g ≥ g ζ ( L ) . It follows that, given ζ ∈ (0 , g ≥ g ζ ( L ) and all m > x ∈ R d P { Λ L ( x ) is ( m, [ − d, − d + B ))-localizing for H g, ω } ≥ − e − L ζ . (2.7)Combining with Theorem 1.6 we obtain the following theorem. Theorem 2.3.
Let H g, ω be an Anderson model with disorder as above, and chooseexponents as in (1.1) - (1.7) . Then, given B > , let J ( B ) = [ − d, − d + B ) andpick < m ≤ log(1 + B d ) . Then, if L is sufficiently large, for all L ≥ L γ and g ≥ g ζ ( L ) we have inf x ∈ R d P (cid:26) Λ L ( x ) is ( m ∞ , J ∞ ( B ) , ( J ∞ ( B )) L γ ) -localizing for H g, ω (cid:27) ≥ − e − L ξ , (2.8) where A ∞ = A ∞ ( L ) = B ∞ Y k =0 (cid:16) − L − κγ k (cid:17) , J ∞ = J ∞ ( L ) = [ − d, − d + A ∞ ) ,m ∞ = m ∞ ( L ) = m ∞ Y k =0 (cid:16) − C d,m − L − ̺γ k (cid:17) . (2.9) In particular, the conclusions of Theorem 1.7 and Corollary 1.8 hold in the interval J ∞ . Moreover, lim L →∞ A ∞ ( L ) = B and lim L →∞ m ∞ ( L ) = m . Preamble to the eigensystem multiscale analysis
In the sections we introduce notation and prove lemmas that play an impor-tant role in the eigensystem multiscale analysis. H will always denote a discreteSchr¨odinger operator H = − ∆ + V on ℓ ( Z d ).3.1. Subsets, boundaries, etc.
Let Φ ⊂ Θ ⊂ Z d . We set the boundary, exteriorboundary, and interior boundary of Φ relative to Θ, respectively, by ∂ Θ Φ = { ( u, v ) ∈ Φ × (Θ \ Φ) ; | u − v | = 1 } , (3.1) ∂ Θex
Φ = n v ∈ (Θ \ Φ) ; ( u, v ) ∈ ∂ Θ Φ for some u ∈ Φ o ,∂ Θin
Φ = n u ∈ Φ; ( u, v ) ∈ ∂ Θ Φ for some v ∈ Θ \ Φ o . We let R ∂ Θin Φ y = dist (cid:0) y, ∂ Θin Φ (cid:1) for y ∈ Φ . (3.2) Given t ≥
1, we setΦ Θ ,t = { y ∈ Φ; dist ( y, Θ \ Φ) > ⌊ t ⌋} , ∂ Θ ,t in Φ = Φ \ Φ Θ ,t , (3.3) ∂ Θ ,t Φ = ∂ Θ ,t in Φ ∪ ∂ Θex Φ . If Θ = Z d we omit it from the notation, i.e., Φ t = Φ Z d ,t . If Φ = Λ L ( x ), we writeΛ Θ ,tL ( x ) = (Λ L ( x )) Θ ,t .Consider a box Λ L ⊂ Θ ⊂ Z d . Given v ∈ Θ, we let ˆ v ∈ ∂ Θin Λ L be the unique u ∈ ∂ Θin Λ L such that ( u, v ) ∈ ∂ Θ Λ L if v ∈ ∂ Θex Λ L , and set ˆ v = 0 otherwise. For L ≥ (cid:12)(cid:12) ∂ Θin Λ L (cid:12)(cid:12) ≤ (cid:12)(cid:12) ∂ Θex Λ L (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∂ Θ Λ L (cid:12)(cid:12)(cid:12) ≤ s d L d − , where s d = 2 d d. (3.4)If Φ ⊂ Θ ⊂ Z d , H Θ = H Φ ⊕ H Θ \ Φ + Γ ∂ Θ Φ on ℓ (Θ) = ℓ (Φ) ⊕ ℓ (Θ \ Φ),where Γ ∂ Θ Φ ( u, v ) = ( − u, v ) or ( v, u ) ∈ ∂ Θ Φ0 otherwise . (3.5)3.2. Lemmas for energy intervals.Lemma 3.1.
Given t > and λ ∈ R , let F t,λ ( z ) be the entire function given by F t,λ ( z ) = 1 − e − t ( z − λ ) z − λ for z ∈ C \ { λ } and F t,λ ( λ ) = 2 tλ. (3.6) Then, given Φ ⊂ Z d , for all x, y ∈ Φ we have |h δ x , F t,λ ( H Φ ) δ y i| ≤ inf η> √ η + λ e t ( η + λ ) e − ( log ( η d )) | x − y | . (3.7) In particular, if λ ∈ I = ( E − A, E + A ) , where A > and E ∈ R , and < m ≤ log (cid:0) A d (cid:1) , (3.8) it follows that for all x, y ∈ Φ , x = y , we have (cid:12)(cid:12)(cid:12)D δ x , F m | x − y | A ,λ − E ( H Φ − E ) δ y E(cid:12)(cid:12)(cid:12) ≤ A − e − mh I ( λ ) | x − y | . (3.9) Proof.
Given t > λ ∈ R , the function F t,λ ( z ) defined in (3.6) is clearly anentire function. Moreover, given η >
0, if | Im z | ≤ η and c > | F t,λ ( z ) | ≤ e t ( η λ +1 c √ η + λ ≤ t ( η λ c √ η + λ if | z − λ | ≥ c p η + λ c +2) √ η + λ (cid:16) e t ( η λ − (cid:17) η + λ ≤ ( c +2)e t ( η λ √ η + λ if | z − λ | < c p η + λ , (3.10)so we conclude that, taking c = √ − F t,λ,η = sup | Im z |≤ η | F t,λ ( z ) | ≤ ( √ t ( η + λ ) p η + λ . (3.11)Given Φ ⊂ Z d , it follows from [AG, Theorem 3] (note that it applies also for H Φ on ℓ (Φ)), that for all x, y ∈ Φ we have |h δ x , F t,λ ( H Φ ) δ y i| ≤ √ F t,λ,η e − ( log ( η d )) | x − y | (3.12) ≤ √ η + λ e t ( η + λ ) e − ( log ( η d )) | x − y | for all η > . IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 13
To prove (3.9), we take E = 0 by replacing the potential V by V − E , and notethat (3.7) holds for any discrete Schr¨odinger operator H . Now let λ ∈ I = ( − A, A ),where
A >
0, and m as in (3.8), and fix x, y ∈ Φ, x = y . Sincelog (cid:0) η d (cid:1) − mA (cid:0) η + λ (cid:1) = log (cid:0) η d (cid:1) − m (cid:16) η A + 1 (cid:17) + mh I ( λ ) , (3.13)choosing η = A , and using (3.8), we obtainlog (cid:0) A d (cid:1) − mA (cid:0) A + λ (cid:1) = log (cid:0) A d (cid:1) − m + mh IA ( λ ) ≥ mh I ( λ ) , (3.14)so (3.9) follows from (3.7) by taking t = m | x − y | A and η = A . (cid:3) Lemma 3.2.
Let Θ ⊂ Z d , and let ψ : Θ → C be a generalized eigenfunction for H Θ with generalized eigenvalue λ ∈ R . Let Φ ⊂ Θ be a finite set such that λ / ∈ σ ( H Φ ) .Let A > , E ∈ R , I = ( E − A, E + A ) . The following holds for all y ∈ Φ : (i) For all t > we have ψ ( y ) = D e − t ( ( H Φ − E ) − ( λ − E ) ) δ y , ψ E − h F t,λ − E ( H Φ − E ) δ y , Γ ∂ Θ Φ ψ i , (3.15) where Γ ∂ Θ Φ is defined in (3.5) and F t,λ ( z ) is the function defined in (3.6) . (ii) Let < R ≤ R ∂ Θin Φ y and m as in (3.8) . For λ ∈ I it follows that (cid:12)(cid:12)(cid:12)D F mRA ,λ − E ( H Φ − E ) δ y , Γ ∂ Θ Φ ψ E(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ∂ Θ Φ (cid:12)(cid:12)(cid:12) A − e − mh I ( λ ) R | ψ ( v ) | , (3.16) for some v ∈ ∂ Θex Φ .Proof. We take E = 0 by replacing the potential V by V − E . By hypothesis wehave λ / ∈ σ ( H Φ ) and h ( H Θ − λ ) ϕ, ψ i = 0 for all ϕ ∈ ℓ (Φ) , (3.17)so (cid:10) ( H Φ − H Θ )( H Φ − λ ) − ϕ, ψ (cid:11) = h ϕ, ψ i for all ϕ ∈ ℓ (Φ) . (3.18)It follows that for all y ∈ Φ and t > ψ ( y ) = h δ y , ψ i (3.19)= D e − t ( H − λ ) δ y , ψ E + D ( H Φ − H Θ )( H Φ − λ ) − (cid:16) − e − t ( H − λ ) (cid:17) δ y , ψ E = D e − t ( H − λ ) δ y , ψ E − h F t,λ ( H Φ ) δ y , Γ ∂ Θ Φ ψ i , where Γ ∂ Θ Φ is defined in (3.5) and the function F t,λ ( z ) is defined in (3.6).Let 0 < R ≤ R ∂ Θin Φ y , m as in (3.8), and assume λ ∈ I = ( − A, A ), A >
0. Recalling(3.5), (3.16) follows from (3.9). (cid:3)
Lemma 3.3.
Let Φ ⊂ Z d , I = ( E − A, E + A ) , where A > and E ∈ R , and λ ∈ I .Then for all t > we have (cid:13)(cid:13)(cid:13) e − t ( ( H Φ − E ) − ( λ − E ) ) χ R \ I ( H Φ ) (cid:13)(cid:13)(cid:13) ≤ e − tA h I ( λ ) . (3.20) Proof.
We have (cid:13)(cid:13)(cid:13) e − t ( ( H Φ − E ) − ( λ − E ) ) χ R \ I ( H Φ ) (cid:13)(cid:13)(cid:13) ≤ e − t ( A − ( λ − E ) ) = e − tA h I ( λ ) . (3.21) (cid:3) Lemmas for the multiscale analysis.
Let I = ( E − A, E + A ) with E ∈ R and A >
0, and fix a constant m − >
0. When we state that a box Λ ℓ is ( m, I )-localizing we always assume0 < m − ℓ − κ ′ ≤ m ≤ log (cid:0) A d (cid:1) . (3.22)We also introduce the following notation: • Given Θ ⊂ Z d and J ⊂ R , we set σ J ( H Θ ) = σ ( H Θ ) ∩ J . • Let Λ ℓ ⊂ Θ ⊂ Z d be an ( m, I )-localizing box with an ( m, I )-localizedeigensystem { ( ϕ ν , ν ) } ν ∈ σ ( H Λ ℓ ) , and let t >
0. Then, for J ⊂ I we set σ Θ ,tJ ( H Λ ℓ ) = n ν ∈ σ J ( H Λ ℓ ); x ν ∈ Λ Θ ,tℓ o . (3.23)The following lemmas plays an important role in our multiscale analysis. Inparticular, the role of the modulating function h I becomes transparent in the proofof Lemma 3.4.3.3.1. Localizing boxes.
Lemma 3.4.
Let ψ : Θ ⊂ Z d → C be a generalized eigenfunction for H Θ withgeneralized eigenvalue λ ∈ I ℓ . Consider a box Λ ℓ ⊂ Θ such that Λ ℓ is ( m, I ) -localizing with an ( m, I ) -localized eigensystem { ϕ ν , ν } ν ∈ σ ( H Λ ℓ ) . Suppose | λ − ν | ≥ e − L β for all ν ∈ σ Θ ,ℓ τ I ( H Λ ℓ ) . (3.24) Then for ℓ sufficiently large we have: (i) If y ∈ Λ Θ , ℓ τ ℓ we have | ψ ( y ) | ≤ e − m h I ( λ ) ℓ τ | ψ ( v ) | for some v ∈ ∂ Θ , ℓ τ Λ ℓ , (3.25) where m = m ( ℓ ) ≥ m (cid:16) − C d,m − ℓ − ( τ − γβ − κ − κ ′ ) (cid:17) . (3.26)(ii) If y ∈ Λ Θ ,ℓ e τ ℓ , we have | ψ ( y ) | ≤ e − m h I ( λ ) R ∂ ΘinΛ ℓy | ψ ( v ) | for some v ∈ ∂ Θ , ℓ τ Λ ℓ , (3.27) where m = m ( ℓ ) ≥ m (cid:16) − C d,m − ℓ − ( − τ ) (cid:17) . (3.28)Lemma 3.4 resembles [EK, Lemma 3.5], but there are important differences.The box Λ ℓ ⊂ Θ is ( m, I )-localizing, and hence we only have decay for eigenfunc-tions with eigenvalues in I . Thus we can only use (3.24) for ν ∈ σ Θ ,ℓ τ I ( H Λ ℓ ). Tocompensate, we take λ ∈ I ℓ , and use Lemmas 3.2 and 3.3. Proof of Lemma 3.4.
We take E = 0 by replacing the potential V by V − E . Given y ∈ Λ, we write ψ ( y ) as in (3.15).Setting P I = χ I ( H Λ ℓ ) and ¯ P I = 1 − P I , we have D e − t ( H − λ ) δ y , ψ E = D e − t ( H − λ ) P I δ y , ψ E + D e − t ( H − λ ) ¯ P I δ y , ψ E . (3.29)It follows from Lemma 3.3 that (cid:12)(cid:12)(cid:12)D e − t ( H − λ ) ¯ P I δ y , ψ E(cid:12)(cid:12)(cid:12) ≤ k χ Λ ψ k (cid:13)(cid:13)(cid:13) e − t ( H − λ ) ¯ P I (cid:13)(cid:13)(cid:13) ≤ ( ℓ + 1) d e − tA h I ( λ ) | ψ ( v ) | , (3.30) IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 15 for some v ∈ Λ.We have D e − t ( H − λ ) P I δ y , ψ E = X µ ∈ σ I ( H Λ ) e − t ( µ − λ ) ϕ µ ( y ) h ϕ µ , ψ i . (3.31)Let y ∈ Λ Θ , ℓ τ ℓ . For µ ∈ σ I ( H Λ ) we have, as shown in [EK, Eqs. (3.37) and(3.39)], | ϕ µ ( y ) h ϕ µ , ψ i| ≤ ℓ d e L β e − mh I ( µ ) ℓ τ | ψ ( v ) | for some v ∈ Λ ℓ ∪ ∂ Θex Λ ℓ . (3.32)It follows thate − t ( µ − λ ) | ϕ µ ( y ) h ϕ µ , ψ i| ≤ ℓ d e L β e − t ( µ − λ ) e − mh I ( µ ) ℓ τ | ψ ( v ) | . (3.33)We now take t = mℓ τ A = ⇒ e − t ( µ − λ ) e − mh I ( µ ) ℓ τ = e − mh I ( λ ) ℓ τ for µ ∈ I, (3.34)obtaining (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) e − mℓ τ A ( H − λ ) P I δ y , ψ (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ℓ d ( ℓ + 1) d e L β e − mh I ( λ ) ℓ τ | ψ ( v ) | (3.35) ≤ e L β e − mh I ( λ ) ℓ τ | ψ ( v ) | , for some v ∈ Λ ℓ ∪ ∂ Θex Λ ℓ . Combining (3.29), (3.30) and (3.35) yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) e − mℓ τ A ( H − λ ) δ y , ψ (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L β e − mh I ( λ ) ℓ τ | ψ ( v ) | , (3.36)for some v ∈ Λ ℓ ∪ ∂ Θex Λ ℓ .Using (3.16), noting y ∈ Λ Θ , ℓ τ ℓ implies R ∂ Θin Λ ℓ y ≥ ℓ τ − > ℓ τ , we get (cid:12)(cid:12)(cid:12)D F mℓτA ,λ ( H Λ ) δ y , Γ ∂ Θ Λ ψ E(cid:12)(cid:12)(cid:12) ≤ s d ℓ d − A − e − mh I ( λ ) ℓ τ | ψ ( v ) | , (3.37)for some v ∈ ∂ Θex
Λ.Combining (3.36) and (3.37), and using (3.22), we conclude that | ψ ( y ) | ≤ C d,m − ℓ κ ′ e L β e − mh I ( λ ) ℓ τ | ψ ( v ) | ≤ e − m h I ( λ ) ℓ τ | ψ ( v ) | , (3.38)for some v ∈ Λ ℓ ∪ ∂ Θex Λ ℓ where, using λ ∈ I ℓ , m ≥ m (cid:16) − C d,m − ℓ − ( τ − γβ − κ − κ ′ ) (cid:17) . (3.39)By repeating the argument as many times a necessary we can get v ∈ ∂ Θ , ℓ τ Λ ℓ .This proves part (i).To prove part (ii), let y ∈ Λ Θ ,ℓ e τ ℓ , so R ∂ Θin Λ ℓ y ≥ ℓ e τ . We proceed as before, butreplace (3.32) by the following estimate. For µ ∈ σ Θ ,tI ( H Λ ℓ ) and v ′ ∈ ∂ Θin Λ ℓ , wehave, as in [EK, Eq. (3.41)], | ϕ µ ( y ) ϕ µ ( v ′ ) | ≤ e − m ′ h I ( µ ) R ∂ ΘinΛ ℓy with m ′ ≥ m (1 − ℓ τ − ) , (3.40)so, as in [EK, Eq. (3.44)], | ϕ µ ( y ) h ϕ µ , ψ i| ≤ L β s d ℓ d − e − m ′ h I ( µ ) R Θ y | ψ ( v ) | ≤ e L β e − m ′ h I ( µ ) R ∂ ΘinΛ ℓy | ψ ( v ) | , (3.41) for some v ∈ ∂ Θex Λ ℓ . If µ ∈ σ I ( H Λ ℓ ) with x µ ∈ ∂ Θ ,ℓ τ in Λ ℓ , we have k x µ − y k ≥ R ∂ Θin Λ ℓ y − ℓ τ ≥ R ∂ Θin Λ ℓ y (cid:16) − ℓ τ − e τ (cid:17) = R ∂ Θin Λ ℓ y (cid:16) − ℓ τ − (cid:17) , (3.42)so | ϕ ν ( y ) h ϕ ν , ψ i| ≤ e − mh I ( µ ) k x µ − y k k χ Λ ψ k (3.43) ≤ e − mh I ( µ ) R ∂ ΘinΛ ℓy (cid:18) − ℓ τ − (cid:19) ( ℓ + 1) d | ψ ( v ) | ≤ ( ℓ + 1) d e − m ′ h I ( µ ) R ∂ ΘinΛ ℓy | ψ ( v ) | , for some v ∈ Λ, where m ′ is given in (3.40). It follows that for all µ ∈ σ I ( H Λ ) wehave e − t ( µ − λ ) | ϕ µ ( y ) h ϕ µ , ψ i| ≤ e L β e − t ( µ − λ ) e − m ′ h I ( µ ) R ∂ ΘinΛ ℓy | ψ ( v ) | , (3.44)for some v ∈ Λ ∪ ∈ ∂ Θex
Λ.We now take t = m ′ R ∂ ΘinΛ ℓy A = ⇒ e − t ( µ − λ ) e − m ′ h I ( µ ) ℓ τ = e − m ′ h I ( λ ) ℓ τ for µ ∈ I, (3.45)obtaining (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* e − m ′ R ∂ ΘinΛ ℓy A ( H − λ ) P I δ y , ψ +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( ℓ + 1) d e L β e − m ′ h I ( λ ) R ∂ ΘinΛ ℓy | ψ ( v ) | (3.46) ≤ e L β e − m ′ h I ( λ ) R ∂ ΘinΛ ℓy | ψ ( v ) | , for some v ∈ Λ ℓ ∪ ∂ Θex Λ ℓ . Combining (3.29), (3.30) and (3.46) yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* e − m ′ R ∂ ΘinΛ ℓy A ( H − λ ) δ y , ψ +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L β e − m ′ h I ( λ ) R ∂ ΘinΛ ℓy | ψ ( v ) | , (3.47)for some v ∈ Λ ℓ ∪ ∂ Θex Λ ℓ .Using (3.16) (with m = m ′ ), we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* F m ′ R∂ ΘinΛ ℓyA ,λ ( H Λ ) δ y , Γ ∂ Θ Λ ψ +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ s d ℓ d − A − e − m ′ h I ( λ ) R ∂ ΘinΛ ℓy | ψ ( v ) | , (3.48)for some v ∈ ∂ Θex
Λ. We conclude from (3.47) and (3.48) that | ψ ( y ) | ≤ C d,m − ℓ κ ′ e L β e − m ′ h I ( λ ) R ∂ ΘinΛ ℓy | ψ ( v ) | ≤ e − m h I ( λ ) R ∂ ΘinΛ ℓy | ψ ( v ) | , (3.49)for some v ∈ Λ ℓ ∪ ∂ Θex Λ ℓ where, using h I ( λ ) ≥ ℓ − κ since λ ∈ I ℓ , we have m ≥ m (cid:16) − C d,m − ℓ − min { e τ − γβ − κ − κ ′ , − τ } (cid:17) = m (cid:16) − C d,m − ℓ − ( − τ ) (cid:17) . (3.50)If v / ∈ ∂ Θ , ℓ τ Λ ℓ , we can apply (3.25) repeatedly until we get (3.49) with v ∈ ∂ Θ , ℓ τ Λ ℓ . (cid:3) Lemma 3.5.
Let the finite set Θ ⊂ Z d be L -level spacing for H , and let { ( ψ λ , λ ) } λ ∈ σ ( H Θ ) be an eigensystem for H Θ .Then the following holds for sufficiently large L : (i) Let Λ ℓ ⊂ Θ be an ( m, I ) -localizing box with an ( m, I ) -localized eigensystem { ( ϕ λ , λ ) } λ ∈ σ ( H Λ ℓ ) . IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 17 (a)
There exists an injection λ ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ) e λ ∈ σ ( H Θ ) , (3.51) such that for all λ ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ) we have (cid:12)(cid:12)(cid:12)e λ − λ (cid:12)(cid:12)(cid:12) ≤ e − m h I ( λ ) ℓ τ , with m = m ( ℓ ) ≥ m (cid:16) − C d,m − log ℓℓ τ − κ − κ ′ (cid:17) , (3.52) and, redefining ϕ λ so (cid:10) ψ e λ , ϕ λ (cid:11) > , (cid:13)(cid:13) ψ e λ − ϕ λ (cid:13)(cid:13) ≤ − m h I ( λ ) ℓ τ e L β . (3.53)(b) Let σ { Λ ℓ } ( H Θ ) := ne λ ; λ ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ) o . (3.54) Then for ν ∈ σ { Λ ℓ } ( H Θ ) we have | ψ ν ( y ) | ≤ − m (2 ℓ ) − κ ℓ τ e L β for all y ∈ Θ \ Λ ℓ . (3.55)(c) If ν ∈ σ I ℓ ( H Θ ) \ σ { Λ ℓ } ( H Θ ) , we have | ν − λ | ≥ e − L β for all λ ∈ σ Θ ,ℓ τ I ( H Λ ℓ ) , (3.56) and | ψ ν ( y ) | ≤ e − m h I ( ν ) ℓ τ for y ∈ Λ Θ , ℓ τ ℓ , with m = m ( ℓ ) as in (3.26) . (3.57) Moreover, if y ∈ Λ Θ ,ℓ e τ ℓ we have | ψ ν ( y ) | ≤ e − m h I ( ν ) R ∂ ΘinΛ ℓy | ψ λ ( y ) | for some y ∈ ∂ Θ , ℓ τ Λ ℓ , (3.58) with m = m ( ℓ ) as is in (3.28) . (ii) Let { Λ ℓ ( a ) } a ∈G , where G ⊂ R d and Λ ℓ ( a ) ⊂ Θ for all a ∈ G , be a collectionof ( m, I ) -localizing boxes with ( m, I ) -localized eigensystems (cid:8) ( ϕ λ ( a ) , λ ( a ) ) (cid:9) λ ( a ) ∈ σ ( H Λ ℓ ( a )) , and set E Θ G ( λ ) = n λ ( a ) ; a ∈ G , λ ( a ) ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ( a )) , e λ ( a ) = λ o for λ ∈ σ ( H Θ ) , (3.59) σ G ( H Θ ) = (cid:8) λ ∈ σ ( H Θ ); E Θ G ( λ ) = ∅ (cid:9) = S a ∈G σ { Λ ℓ ( a ) } ( H Θ ) . (a) Let a, b ∈ G , a = b , Then, for λ ( a ) ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ( a )) and λ ( b ) ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ( b )) , λ ( a ) , λ ( b ) ∈ E Θ G ( λ ) = ⇒ k x λ ( a ) − x λ ( b ) k < ℓ τ . (3.60) As a consequence, Λ ℓ ( a ) ∩ Λ ℓ ( b ) = ∅ = ⇒ σ { Λ ℓ ( a ) } ( H Θ ) ∩ σ { Λ ℓ ( b ) } ( H Θ ) = ∅ . (3.61)(b) If λ ∈ σ G ( H Θ ) , we have | ψ λ ( y ) | ≤ − m (2 ℓ ) − κ ℓ τ e L β for all y ∈ Θ \ Θ G , where Θ G := [ a ∈G Λ ℓ ( a ) . (3.62)(c) If λ ∈ σ I ℓ ( H Θ ) \ σ G ( H Θ ) , we have | ψ λ ( y ) | ≤ e − m h I ( λ ) ℓ τ for all y ∈ Θ G ,τ := [ a ∈G Λ Θ , ℓ τ ℓ ( a ) . (3.63) Proof.
Let Λ ℓ ⊂ Θ be be an ( m, I )-localizing box with an ( m, I )-localized eigensys-tem { ( ϕ λ , λ ) } λ ∈ σ ( H Λ ℓ ) . Given λ ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ), it follows from [EK, Eq. (3.10) inLemma 3.2] that dist ( λ, σ ( H Θ )) ≤ √ s d ℓ d − e − mh I ( λ ) ℓ τ , (3.64)so the existence of e λ ∈ σ ( H Θ ) satisfying (3.52) follows. Uniqueness follows fromthe fact that Θ is L -level spacing and γβ < τ . In addition, note that e λ = e ν if λ, ν ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ), λ = ν , because in this case we have (cid:12)(cid:12)(cid:12)e λ − e ν (cid:12)(cid:12)(cid:12) ≥ | λ − ν | − (cid:12)(cid:12)(cid:12)e λ − λ (cid:12)(cid:12)(cid:12) − | e ν − ν | ≥ e − ℓ β − − m (2 ℓ ) − κ ℓ τ ≥ e − ℓ β , (3.65)as Λ ℓ ( a ) is level spacing for H , and κ + β < τ . Moreover, it follows from [EK,Lemma 3.3] that, after multiplying ϕ λ by a phase factor if necessary to get so (cid:10) ψ e λ , ϕ λ (cid:11) >
0, we have (3.53).If ν ∈ σ { Λ ℓ } ( H Θ ), we have ν = e λ for some λ ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ), so (3.55) follows from(3.53) as ϕ λ ( y ) = 0 for all y ∈ Θ \ Λ ℓ ( a ).Let ν ∈ σ I ℓ ( H Θ ) \ σ { Λ ℓ } ( H Θ ). Then for all λ ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ) we have | ν − λ | ≥ (cid:12)(cid:12)(cid:12) ν − e λ (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)e λ − λ (cid:12)(cid:12)(cid:12) ≥ e − L β − e − m h I ( λ ) ℓ τ (3.66) ≥ e − L β − e − m (2 ℓ ) − κ ℓ τ ≥ e − L β , since Θ is L -level spacing for H , we have (3.52), and κ + γβ < τ . Thus | ν − λ | ≥ e − L β for all λ ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ) . (3.67)Since ν ∈ I ℓ , we actually have (3.56). Thus (3.57) follows from Lemma 3.4(i) and k ψ ν k = 1, and (3.58) follows from Lemma 3.4(ii).Now let { Λ ℓ ( a ) } a ∈G , where G ⊂ R d and Λ ℓ ( a ) ⊂ Θ for all a ∈ G , be a collection of( m, I )-localizing boxes with ( m, I )-localized eigensystems (cid:8) ( ϕ λ ( a ) , λ ( a ) ) (cid:9) λ ( a ) ∈ σ ( H Λ ℓ ( a )) .Let λ ∈ σ ( H Θ ), a, b ∈ G , a = b , λ ( a ) ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ( a )) and λ ( b ) ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ( b )).Suppose λ ( a ) , λ ( b ) ∈ E Θ G ( λ ), where E Θ G ( λ ) is given in (3.59). It then follows from(3.53) that k ϕ λ ( a ) − ϕ λ ( b ) k ≤ − m (2 ℓ ) − κ ℓ τ e L β , (3.68)so |h ϕ λ ( a ) , ϕ λ ( b ) i| ≥ ℜ D ϕ ( a ) x , ϕ ( b ) y E ≥ − − m (2 ℓ ) − κ ℓ τ e L β . (3.69)On the other hand, it follows from (1.11) that k x λ ( a ) − x λ ( b ) k ≥ ℓ τ = ⇒ (cid:12)(cid:12)(cid:12)D ϕ ( a ) x , ϕ ( b ) y E(cid:12)(cid:12)(cid:12) ≤ ( ℓ + 1) d e − m (2 ℓ ) − κ ℓ τ . (3.70)Combining (3.69) and (3.70) we conclude that λ ( a ) , λ ( b ) ∈ E Θ G ( λ ) = ⇒ k x λ ( a ) − x λ ( b ) k < ℓ τ . (3.71)To prove (3.61), let a, b ∈ G , a = b . If Λ ℓ ( a ) ∩ Λ ℓ ( b ) = ∅ , we have that λ ( a ) ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ( a )) and λ ( b ) ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ( b )) = ⇒ k x λ ( a ) − x λ ( b ) k ≥ ℓ τ , (3.72)so it follows from (3.60) that σ { Λ ℓ ( a ) } ( H Θ ) ∩ σ { Λ ℓ ( b ) } ( H Θ ) = ∅ .Parts (ii)(b) and (ii)(c) are immediate consequence of parts (i)(b) and (i)(c),respectively. (cid:3) IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 19
Buffered subsets.
In the multiscale analysis we will need to consider boxesΛ ℓ ⊂ Λ L that are not ( m, I )-localizing for H . Instead of studying eigensystemsfor such boxes, we will surround them with a buffer of ( m, I )-localizing boxes andstudy eigensystems for the augmented subset. Definition 3.6.
We call Υ ⊂ Λ L an ( m, I ) -buffered subset of the box Λ L if thefollowing holds: (i) Υ is a connected set in Z d of the form Υ = J [ j =1 Λ R j ( a j ) ∩ Λ L , (3.73) where J ∈ N , a , a , . . . , a J ∈ Λ R L , and ℓ ≤ R j ≤ L for j = 1 , , . . . , J . (ii) Υ is L -level spacing for H . (iii) There exists G Υ ⊂ Λ R L such that: (a) For all a ∈ G Υ we have Λ ℓ ( a ) ⊂ Υ , and Λ ℓ ( a ) is an ( m, I ) -localizingbox for H . (b) For all y ∈ ∂ Λ L in Υ there exists a y ∈ G Υ such that y ∈ Λ Υ , ℓ τ ℓ ( a y ) .In this case we set b Υ = [ a ∈G Υ Λ ℓ ( a ) , b Υ τ = [ a ∈G Υ Λ Υ , ℓ τ ℓ ( a ) , b Υ = Υ \ b Υ , and b Υ τ = Υ \ b Υ τ . (3.74) ( b Υ = Υ G Υ and b Υ τ = Υ G Υ ,τ in the notation of Lemma 3.5.) The set b Υ τ ⊃ ∂ Λ L in Υ is a localizing buffer between b Υ and Λ L \ Υ, as shown inthe following lemma.
Lemma 3.7.
Let Υ be an ( m, I ) -buffered subset of Λ L , and let { ( ψ ν , ν ) } ν ∈ σ ( H Υ ) be an eigensystem for H Υ . Let G = G Υ and set σ B ( H Υ ) = σ I ℓ ( H Υ ) \ σ G ( H Υ ) , (3.75) where σ G ( H Υ ) is as in (3.59) . Then the following holds for sufficiently large L : (i) For all ν ∈ σ B ( H Υ ) we have | ψ ν ( y ) | ≤ e − m h I ( ν ) ℓ τ for all y ∈ b Υ τ , with m = m ( ℓ ) as in (3.26) . (3.76) . (ii) Let Λ L be level spacing for H , and let { ( φ λ , λ ) } λ ∈ σ ( H Λ L ) be an eigensystemfor H Λ L . There exists an injection ν ∈ σ B ( H Υ ) e ν ∈ σ ( H Λ L ) \ σ G ( H Λ L ) , (3.77) such that for ν ∈ σ B ( H Υ ) we have | e ν − ν | ≤ e − m ℓ τ − κ , with m = m ( ℓ ) ≥ m (cid:16) − C d,m − ℓ − ( τ − γβ − κ − κ ′ ) (cid:17) , (3.78) and, redefining ψ ν so h φ e ν , ψ ν i > , k φ e ν − ψ ν k ≤ − m ℓ τ − κ e L β . (3.79) Proof.
Part (i) follows immediately from Lemma 3.5(ii)(c).Now let Λ L be level spacing for H , and let { ( φ λ , λ ) } λ ∈ σ ( H Λ L ) be an eigensystemfor H Λ L . It follows from [EK, Eq. (3.11) in Lemma 3.2] that for ν ∈ σ B ( H Υ ) wehave k ( H Λ L − ν ) ψ ν k ≤ (2 d − (cid:12)(cid:12) ∂ Λ L ex Υ (cid:12)(cid:12) (cid:13)(cid:13)(cid:13) ψ ν χ ∂ Λ L in Υ (cid:13)(cid:13)(cid:13) ∞ ≤ (2 d − L d e − m h I ( ν ) ℓ τ ≤ (2 d − L d e − m ℓ − κ ℓ τ ≤ e − m ℓ τ − κ , (3.80)where we used ∂ Λ L in Υ ⊂ b Υ τ and (3.76), and m is given in (3.78). Since Λ L and Υare L -level spacing for H , the map in (3.77) is a well defined injection into σ ( H Λ L ),and (3.79) follows from (3.78) and [EK, Lemma 3.3].To finish the proof we must show that e ν / ∈ σ G ( H Λ L ) for all ν ∈ σ B ( H Υ ). Suppose e ν ∈ σ G ( H Λ L ) for some ν ∈ σ B ( H Υ ). Then there is a ∈ G and λ ( a ) ∈ σ Θ ,ℓ τ I ℓ ( H Λ ℓ ( a ))such that λ ( a ) ∈ E Λ L G ( e ν ). On the other hand, it follows from Lemma 3.5(i)(a) that λ ( a ) ∈ E Υ G ( λ ) for some λ ∈ σ G ( H Υ ). We conclude from (3.53) and (3.79) that √ k ψ λ − ψ ν k ≤ k ψ λ − ϕ λ ( a ) k + k ϕ λ ( a ) − φ e ν k + k φ e ν − ψ ν k (3.81) ≤ − m (2 ℓ ) − κ ℓ τ e L β + 2e − m ℓ τ − κ e L β < , a contradiction. (cid:3) Lemma 3.8.
Let Λ L = Λ L ( x ) , x ∈ R d . Let Υ be an ( m, I ) -buffered subset of Λ L . Let G = G Υ , and for ν ∈ σ ( H Υ ) set E Λ L G ( ν ) = n λ ( a ) ; a ∈ G , λ ( a ) ∈ σ Λ L ,ℓ τ I ℓ ( H Λ ℓ ( a )) , e λ ( a ) = ν o ⊂ E Υ G ( ν ) ,σ Λ L G ( H Υ ) = n ν ∈ σ ( H Υ ); E Λ L G ( ν ) = ∅ o ⊂ σ G ( H Υ ) . (3.82) The following holds for sufficiently large L : (i) Let ( ψ, λ ) be an eigenpair for H Λ L such that λ ∈ I ℓ and | λ − ν | ≥ e − L β for all ν ∈ σ Λ L G ( H Υ ) ∪ σ B ( H Υ ) . (3.83) Then for all y ∈ Υ Λ L , ℓ τ we have | ψ ( y ) | ≤ e − m h Iℓ ( λ ) ℓ τ | ψ ( v ) | for some v ∈ ∂ Λ L , ℓ τ Υ , (3.84) where m = m ( ℓ ) ≥ m (cid:16) − C d,m − ℓ − min { κ,τ − γβ − κ − κ ′ } (cid:17) . (3.85)(ii) Let Λ L be level spacing for H , let { ( ψ λ , λ ) } λ ∈ σ ( H Λ L ) be an eigensystem for H Λ L , recall (3.77) , and set σ Υ ( H Λ L ) = { e ν ; ν ∈ σ B ( H Υ ) } ⊂ σ ( H Λ L ) \ σ G ( H Λ L ) . (3.86) Then for all λ ∈ σ I ℓ ( H Λ L ) \ ( σ G ( H Λ L ) ∪ σ Υ ( H Λ L )) , the condition (3.83) is satisfied, and ψ λ satisfies (3.84) . IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 21
Proof.
To prove part (i), we take E = 0 by replacing the potential V by V − E .Let ( ψ, λ ) be an eigenpair for H Λ L satisfying (3.83). Given y ∈ Υ, we write ψ ( y ) asin (3.15). We set P = χ I ℓ ( H Υ ) and ¯ P = 1 − P I ℓ . We use Lemma 3.3 with Φ = Υand J = I ℓ .To estimate D e − t ( H − λ ) P δ y , ψ E , let { ( ϑ ν , ν ) } ν ∈ σ ( H Υ ) be an eigensystem for H Υ .For each ν ∈ σ G ( H Υ ) we fix λ ( a ν ) ∈ E Υ G ( ν ), where a ν ∈ G , λ ( a ν ) ∈ σ Λ L ,ℓ τ I ℓ ( H Λ ℓ ( a ν )),picking λ ( a ν ) ∈ E Λ L G ( ν ) if ν ∈ σ Λ L G ( H Υ ), so x λ ( aν ) ∈ Λ Λ L ,ℓ τ ℓ ( a ν ). If ν ∈ σ G ( H Υ ) \ σ Λ L G ( H Υ ) we have x λ ( aν ) ∈ Λ Υ ,ℓ τ ℓ ( a ν ) \ Λ Λ L ,ℓ τ ℓ ( a ν ).Given J ⊂ R , we set σ G ,J ( H Υ ) = σ G ( H Υ ) ∩ J , σ Λ L G ,J ( H Υ ) = σ Λ L G ( H Υ ) ∩ J . Wehave D e − t ( H − λ ) P δ y , ψ E = X ν ∈ σ Iℓ (Υ) e − t ( ν − λ ) ϑ ν ( y ) h ϑ ν , ψ i (3.87)= X ν ∈ σ Λ L G ,Iℓ ( H Υ ) ∪ σ B ( H Υ ) e − t ( ν − λ ) ϑ ν ( y ) h ϑ ν , ψ i + X ν ∈ σ G ,Iℓ ( H Υ ) \ σ Λ L G ,Iℓ ( H Υ ) e − t ( ν − λ ) ϑ ν ( y ) h ϑ ν , ψ i . If ν ∈ σ Λ L G ( H Υ ) ∪ σ B ( H Υ ) we have h ϑ ν , ψ i = ( λ − ν ) − h ϑ ν , ( H Λ L − ν ) ψ i = ( λ − ν ) − h ( H Λ L − ν ) ϑ ν , ψ i . (3.88)It follows from (3.83) and [EK, Eq. (3.10) in Lemma 3.2] that | ϑ ν ( y ) h ϑ ν , ψ i| ≤ L β | ϑ ν ( y ) | X v ∈ ∂ Λ L ex Υ P v ′ ∈ ∂ Λ L in Υ | v ′ − v | =1 | ϑ ν ( v ′ ) | | ψ ( v ) | (3.89) ≤ L d e L β ( d max u ∈ ∂ Λ L in Υ | ϑ ν ( u ) | ) | ψ ( v ) | for some v ∈ ∂ Λ L ex Υ . If ν ∈ σ B ( H Υ ) it follows from (3.76) thatmax u ∈ ∂ Λ L in Υ | ϑ ν ( u ) | ≤ e − m h I ( ν ) ℓ τ . (3.90)If ν ∈ σ Λ L G ,J ( H Υ ), it follows from (3.53) and (1.11) thatmax u ∈ ∂ Λ L in Υ | ϑ ν ( u ) | ≤ max u ∈ ∂ Λ L in Υ (cid:16)(cid:12)(cid:12)(cid:12) ϑ ν ( u ) − ϕ ( a ν ) ( u ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ϕ ( a ν ) ( u ) (cid:12)(cid:12)(cid:12)(cid:17) (3.91) ≤ − m h I ( λ ( aν ) ) ℓ τ e L β + e − mh I ( λ ( aν ) ) ℓ τ ≤ − m h I ( λ ( aν ) ) ℓ τ e L β ≤ − m ′ h I ( ν ) ℓ τ e L β , where m ′ ≥ m (1 − e − C m − ℓ τ − κ − κ ′ ) , where we used (3.52). It follows that, with m ′′ = min { m ′ , m } ≥ m (cid:16) − C d,m − ℓ − ( τ − γβ − κ − κ ′ ) (cid:17) , (3.92)for all ν ∈ σ Λ L G ,J ( H Υ ) ∪ σ B ( H Υ ) we have | ϑ ν ( y ) h ϑ ν , ψ i| ≤ e L β e − m ′′ h I ( ν ) ℓ τ | ψ ( v ) | for some v ∈ ∂ Λ L ex Υ . (3.93) Picking t = m ′′ ℓ τ A , for all ν ∈ σ Λ L G ,J ( H Υ ) ∪ σ B ( H Υ ) we gete − t ( ν − λ ) | ϑ ν ( y ) h ϑ ν , ψ i| ≤ e L β e − m ′′ h I ( λ ) ℓ τ | ψ ( v ) | for some v ∈ ∂ Λ L ex Υ , (3.94)so (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ν ∈ σ Λ L G ,J ( H Υ ) ∪ σ B ( H Υ ) e − t ( ν − λ ) ϑ ν ( y ) h ϑ ν , ψ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( L + 1) d e L β e − m ′′ h I ( λ ) ℓ τ | ψ ( v ) | , (3.95)for some v ∈ ∂ Λ L ex Υ.Now let ν ∈ σ G ,J ( H Υ ) \ σ Λ L G ,J ( H Υ ). In this case we have x λ ( aν ) ∈ Λ Υ ,ℓ τ ℓ ( a ν ) \ Λ Λ L ,ℓ τ ℓ ( a ν ), so we havedist ( x λ ( aν ) , Υ \ Λ ℓ ( a ν )) > ℓ τ and dist ( x λ ( aν ) , Λ L \ Λ ℓ ( a ν )) ≤ ℓ τ , (3.96)so there is u ∈ Λ L \ Υ such that k x λ ( aν ) − u k ≤ ℓ τ . We now assume y ∈ Υ Λ L , ℓ τ ,so we have k y − u k > ℓ τ . We conclude that | x λ ( aν ) − y | ≥ k y − u k − k x λ ( aν ) − u k > ℓ τ − ℓ τ = ℓ τ . (3.97)Thus | ϑ ν ( y ) | ≤ | ϑ ν ( y ) − ϕ λ ( aν ) ( y ) | + | ϕ λ ( aν ) ( y ) | (3.98) ≤ − m h I ( λ ( aν ) ) ℓ τ e L β + e − mh I ( λ ( aν ) ) ℓ τ ≤ − m h I ( λ ( aν ) ) ℓ τ e L β ≤ − m ′ h I ( ν ) ℓ τ e L β , using (3.53), (1.11), and (3.52). It follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ν ∈ σ G ,J ( H Υ ) \ σ Λ L G ,J ( H Υ ) e − t ( ν − λ ) ϑ ν ( y ) h ϑ ν , ψ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L + 1) d e − m ′′ h I ( λ ) ℓ τ e L β | ψ ( v ) | , (3.99)for some v ∈ Υ.Combining (3.87), (3.95) and (3.99), we get for y ∈ Υ Λ L , ⌊ ℓ τ ⌋ that (cid:12)(cid:12)(cid:12)D e − t ( H − λ ) P δ y , ψ E(cid:12)(cid:12)(cid:12) ≤ e L β e − m ′′ h I ( λ ) ℓ τ | ψ ( v ) | , (3.100)for some v ∈ Υ ∪ ∂ Λ L ex Υ.From Lemma 3.3, we get, (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) e − m ′′ ℓτA ( H − λ ) ¯ P δ y , ψ (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( ℓ + 1) d e m ′′ (1 − ℓ − κ ) h Iℓ ( λ ) ℓ τ | ψ ( v ) | , (3.101)for some v ∈ Υ.Combining (3.100) and (3.101), we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) e − m ′′ ℓτA ( H − λ ) δ y , ψ (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L β e − m ′′ (1 − ℓ − κ ) h Iℓ ( λ ) ℓ τ | ψ ( v ) | , (3.102)for some v ∈ Υ ∪ ∂ Λ L ex Υ. IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 23
Using (3.16) (with m = m ′′ , I = I ℓ , R = ℓ τ ), noting y ∈ Υ Λ L , ℓ τ implies R ∂ Λ L in Υ y ≥ ℓ τ − > ℓ τ , we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) F m ′′ ℓτA ,λ ( H Υ ) δ y , Γ ∂ Θ Υ ψ (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L d A − e − m ′′ h I ( λ ) ℓ τ | ψ ( v ) | , (3.103)for some v ∈ ∂ Θex
Υ.Combining (3.102) and (3.103)we get | ψ ( y ) | ≤ C d,m − ℓ κ ′ e L β e − m ′′ (1 − ℓ − κ ) h Iℓ ( λ ) ℓ τ | ψ ( v ) | ≤ e − m h Iℓ ( λ ) ℓ τ | ψ ( v ) | , (3.104)for some v ∈ Υ ∪ ∂ Λ L ex Υ, where m is as in (3.85). Repeating the procedure asmany times as needed, we can require v ∈∈ ∂ Λ L , ℓ τ Υ.Now suppose Λ L is level spacing for H , and let λ ∈ σ I ℓ ( H Λ L ) \ ( σ G ( H Λ L ) ∪ σ Υ ( H Λ L )).If λ / ∈ σ G ( H Λ L ), it follows from Lemma 3.5(i)(c) that (3.56) holds for all a ∈ G .If λ / ∈ σ Υ ( H Λ L ), the argument in (3.66), modified by the use of (3.78) instead of(3.52), using (1.5), gives | λ − ν | ≥ e − L β for all ν ∈ σ B ( H Υ ). Thus we have (3.83),which implies (3.84). (cid:3) Suitable covers of a box.
To perform the multiscale analysis in an efficientway, it is convenient to use a canonical way to cover a box of side L by boxes of side ℓ < L . We will use the idea of suitable covers of a box as in [GK4, Definition 3.12],adapted to the discrete case. Since we will use (3.27) to get decay of eigenfunctionsin scale L from decay in scale ℓ , we will need to make sure R ∂ Λ L in Λ ℓ y ≈ ℓ . We willdo so by ensuring that for all y ∈ Λ L we can find a box Λ ℓ in the cover such that y ∈ Λ ℓ with R ∂ Λ L in Λ ℓ y ≈ ℓ ≥ ℓ − ℓ ς − ς ∈ (0 , ς asin (1.7) for convenience. Definition 3.9.
Fix ς ∈ (0 , . Let Λ L = Λ L ( x ) , x ∈ R d be a box in Z d , and let ℓ < L . A suitable ℓ -cover of Λ L is the collection of boxes C L,ℓ = C L,ℓ ( x ) = { Λ ℓ ( a ) } a ∈ Ξ L,ℓ , (3.105) where Ξ L,ℓ = Ξ
L,ℓ ( x ) := (cid:8) x + ρℓ ς Z d (cid:9) ∩ Λ R L with ρ ∈ (cid:2) , (cid:3) ∩ (cid:8) L − ℓ ℓ ς k ; k ∈ N (cid:9) . (3.106) We call C L,ℓ the suitable ℓ -cover of Λ L if ρ = ρ L,ℓ := max (cid:2) , (cid:3) ∩ (cid:8) L − ℓ ℓ ς k ; k ∈ N (cid:9) . We adapt [GK4, Lemma 3.13] to our context.
Lemma 3.10.
Let ℓ ≤ L . Then for every box Λ L = Λ L ( x ) , x ∈ R d , a suitable ℓ -cover C L,ℓ = C L,ℓ ( x ) satisfies Λ L = [ a ∈ Ξ L,ℓ Λ ℓ ( a ); (3.107) for all b ∈ Λ L there is Λ ( b ) ℓ ∈ C L,ℓ such that b ∈ (cid:16) Λ ( b ) ℓ (cid:17) Λ L , ℓ − ℓς , (3.108) i.e., Λ L = [ a ∈ Ξ L,ℓ Λ Λ L , ℓ − ℓς ℓ ( a ); L,ℓ = (cid:16) L − ℓρℓ ς + 1 (cid:17) d ≤ (cid:0) Lℓ ς (cid:1) d . (3.109) Moreover, given a ∈ x + ρℓ ς Z d and k ∈ N , it follows that Λ (2 kρℓ ς + ℓ ) ( a ) = [ b ∈{ x + ρℓ ς Z d }∩ Λ R (2 kρℓς + ℓ ) ( a ) Λ ℓ ( b ) , (3.110) and { Λ ℓ ( b ) } b ∈{ x + ρℓ ς Z d }∩ Λ R (2 kρℓς + ℓ ) ( a ) is a suitable ℓ -cover of the box Λ (2 kρℓ ς + ℓ ) ( a ) . Note that Λ ( b ) ℓ does not denote a box centered at b , just some box in C L,ℓ ( x )satisfying (3.108). By Λ ( b ) ℓ we will always mean such a box. We will use R ∂ Λ L in Λ ( b ) ℓ b ≥ ℓ − ℓ ς − b ∈ Λ L . (3.111)Note also that ρ ≤ ρ = ρ L,ℓ in for the suitable ℓ -coverfor convenience, so there is no ambiguity in the definition of C L,ℓ ( x ).Suitable covers are convenient for the construction of buffered subsets (see Def-inition 3.6) in the multiscale analysis. We will use the following observation: Remark 3.11.
Let C L,ℓ be a suitable ℓ -cover for the box Λ L , and set k ℓ = k L,ℓ = (cid:4) ρ − ℓ − ς (cid:5) + 1 . (3.112) Then for all a, b ∈ C
L,ℓ we have Λ R ℓ ( a ) ∩ Λ R ℓ ( b ) = ∅ ⇐⇒ k a − b k ≥ k ℓ ρℓ ς . (3.113)3.5. Probability estimate for level spacing.
The eigensystem multiscale anal-ysis uses a probability estimate of Klein and Molchanov [KlM, Lemma 2], whichwe state as in [EK, Lemma 2.1]. If J ⊂ R , we set diam J = sup s,t ∈ J | s − t | . Lemma 3.12.
Let H ω be an Anderson model as in Definition 1.15. Let Θ ⊂ Z d and L > . Then P { Θ is L -level spacing for H ω } ≥ − Y µ e − (2 α − L β | Θ | . (3.114) where Y µ = 2 α − e K (diam supp µ + 2 d + 1) , (3.115) with e K = K if α = 1 and e K = 8 K if α ∈ ( , .In the special case of a box Λ L , we have P { Λ L is level spacing for H } ≥ − Y µ ( L + 1) d e − (2 α − L β . (3.116)4. Eigensystem multiscale analysis
In this section we fix an Anderson model H ω and prove Theorem 1.6. Note that ̺ is given in (1.6). Proposition 4.1.
Fix m − > . There exists a a finite scale L = L ( d, m − ) withthe following property: Suppose for some scale L ≥ L we have inf x ∈ R d P { Λ L ( x ) is ( m , I ) -localizing for H ω } ≥ − e − L ζ , (4.1) where I = ( E − A , E + A ) ⊂ R , with E ∈ R and A > , and m − L − κ ′ ≤ m ≤ log (cid:0) A d (cid:1) . (4.2) IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 25
Set L k +1 = L γk , A k +1 = A k (1 − L − κk ) , and I k +1 = ( E − A k +1 , E + A k +1 ) , for k = 0 , , . . . . Then for all k = 1 , , . . . we have inf x ∈ R d P { Λ L k ( x ) is ( m k , I k , I k − ) -localizing for H ω } ≥ − e − L ζk , (4.3) where m − L − κ ′ k < m k − (cid:0) − C d,m − L − ̺k − (cid:1) ≤ m k < log (cid:0) A k d (cid:1) . (4.4)The proof of Proposition 4.1 relies on the following lemma, the induction stepfor the multiscale analysis. Lemma 4.2.
Fix m − > . Let I = ( E − A, E + A ) ⊂ R , with E ∈ R and A > ,and m > . Suppose for some scale ℓ we have inf x ∈ R d P { Λ ℓ ( x ) is ( m, I ) -localizing for H ω } ≥ − e − ℓ ζ , (4.5) where m − ℓ − κ ′ ≤ m ≤ log (cid:0) A d (cid:1) . (4.6) Then, if ℓ is sufficiently large, we have (recall L = ℓ γ ) inf x ∈ R d P { Λ L ( x ) is ( M, I ℓ , I ) -localizing for H ω } ≥ − e − L ζ , (4.7) where m − L − κ ′ < m (cid:0) − C d,m − ℓ − ̺ (cid:1) ≤ M < log (cid:16) A (1 − ℓ − κ )4 d (cid:17) . (4.8) Proof.
To prove the lemma we proceed as in [EK, Proof of Lemma 4.5], with somemodifications. The crucial estimate (3.27) is a somewhat weaker statement than itscounterpart [EK, Eq. (3.31)]. For this reason we are forced to modify the definitionof an ℓ -cover of a box, and use the version given in Definition 3.9 with ς as in (1.7),which differs from the version given in [EK, Definition 3.10] which has ς = 1. Inparticular, we have (3.113), while in [EK] the corresponding statement holds withthe simpler k a − b k ≥ ρℓ .We assume (4.5) and (4.6) for a scale ℓ . We take Λ L = Λ( x ), where x ∈ R d ,and let C L,ℓ = C L,ℓ ( x ) be the suitable ℓ -cover of Λ L (with ς as in (1.7)). Given a, b ∈ Ξ L,ℓ , we will say that the boxes Λ ℓ ( a ) and Λ ℓ ( b ) are disjoint if and only ifΛ R ℓ ( a ) ∩ Λ R ℓ ( b ) = ∅ , that is, if and only if k a − b k ≥ k ℓ ρℓ ς (see Remark 3.11). Wetake N = N ℓ = j ℓ ( γ − e ζ k (recall (1.3)), and let B N denote the event that thereexist at most N disjoint boxes in C L,ℓ that are not ( m, I )-localizing for H ω . Forsufficiently large ℓ , we have, using (3.109), (4.5), and the fact that events on disjointboxes are independent, that P {B cN } ≤ (cid:0) Lℓ ς (cid:1) ( N +1) d e − ( N +1) ℓ ζ = 2 ( N +1) d ℓ ( γ − ς )( N +1) d e − ( N +1) ℓ ζ < e − L ζ . (4.9)We now fix ω ∈ B N . There exists A N = A N ( ω ) ⊂ Ξ L,ℓ = Ξ
L,ℓ ( x ) such that |A N | ≤ N and k a − b k ≥ k ℓ ρℓ ς if a, b ∈ A N and a = b , with the following property:if a ∈ Ξ L,ℓ with dist( a, A N ) ≥ k ℓ ρℓ ς , so Λ R ℓ ( a ) ∩ Λ R ℓ ( b ) = ∅ for all b ∈ A N , the boxΛ ℓ ( a ) is ( m, I )-localizing for H ω . In other words, a ∈ Ξ L,ℓ \ [ b ∈A N Λ R k ℓ − ρℓ ς ( b ) = ⇒ Λ ℓ ( a ) is ( m, I )-localizing for H ω . (4.10) We want to embed the boxes { Λ ℓ ( b ) } b ∈A N into ( m, I )-buffered subsets of Λ L .To do so, we consider graphs G i = (Ξ L,ℓ , E i ), i = 1 ,
2, both having Ξ
L,ℓ as the setof vertices, with sets of edges given by E = (cid:8) { a, b } ∈ Ξ L,ℓ ; k a − b k ≤ ( k ℓ − ρℓ ς (cid:9) (4.11)= (cid:8) { a, b } ∈ Ξ L,ℓ ; a = b and Λ R ℓ ( a ) ∩ Λ R ℓ ( b ) = ∅ (cid:9) , E = (cid:8) { a, b } ∈ Ξ L,ℓ ; k ℓ ρℓ ς ≤ k a − b k ≤ k ℓ − ρℓ ς (cid:9) = (cid:8) { a, b } ∈ Ξ L,ℓ ; Λ R ℓ ( a ) ∩ Λ R ℓ ( b ) = ∅ and k a − b k ≤ k ℓ − ρℓ ς (cid:9) . Given Ψ ⊂ Ξ L,ℓ , we let Ψ = Ψ ∪ ∂ G ex Ψ, where ∂ G ex Ψ, the exterior boundary of Ψin the graph G , is defined by ∂ G ex Ψ = { a ∈ Ξ L,ℓ \ Ψ; dist( a, Ψ) ≤ ( k ℓ − ρℓ ς } (4.12)= { a ∈ Ξ L,ℓ \ Ψ; ( b, a ) ∈ E for some b ∈ Ψ } . Let Φ ⊂ Ξ L,ℓ be G -connected, so diam Φ ≤ ρℓ ( | Φ | − e Φ = Ξ
L,ℓ ∩ [ a ∈ Φ Λ R (2 ρ +1) ℓ ( a ) = { a ∈ Ξ L,ℓ ; dist( a, Φ) ≤ ρℓ } (4.13)is a G -connected subset of Ξ L,ℓ such thatdiam e Φ ≤ diam Φ + 2 ρℓ ≤ k ℓ − ρℓ ς ( | Φ | − . (4.14)We setΥ (0)Φ = [ a ∈ e Φ Λ ℓ ( a ) and Υ Φ = Υ (0)Φ ∪ [ a ∈ ∂ G e Φ Λ ℓ ( a ) = [ a ∈ e Φ Λ ℓ ( a ) . (4.15)Let { Φ r } Rr =1 = { Φ r ( ω ) } Rr =1 denote the G -connected components of A N (i.e.,connected in the graph G ); we have R ∈ { , , . . . , N } and P Rr =1 | Φ r | = |A N | ≤ N .We conclude that ne Φ r o Rr =1 is a collection of disjoint, G -connected subsets of Ξ L,ℓ ,such that dist( e Φ r , e Φ s ) ≥ k ℓ ρℓ ς if r = s. (4.16)Moreover, it follows from (4.10) that a ∈ G = G ( ω ) = Ξ L,ℓ \ R [ r =1 e Φ r = ⇒ Λ ℓ ( a ) is ( m, I )-localizing for H ω . (4.17)In particular, we conclude that Λ ℓ ( a ) is ( m, I )-localizing for H ω for all a ∈ ∂ G ex e Φ r , r = 1 , , . . . , R .Each Υ r = Υ Φ r , r = 1 , , . . . , R , clearly satisfies all the requirements to be an( m, I )-buffered subset of Λ L with G Υ r = ∂ G ex e Φ r (see Definition 3.6), except thatwe do not know if Υ r is L -level spacing for H ω . (Note that the sets { Υ (0) r } Rr =1 aredisjoint, but the sets { Υ r } Rr =1 are not necessarily disjoint.) Note also that it followsfrom (4.14) thatdiam Υ r ≤ diam e Φ r + ℓ ≤ ( k ℓ − ρℓ ς (3 | Φ r | + 1) + ℓ ≤ ℓ | Φ r | , (4.18) IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 27 so, using (1.4), we have R X r =1 diam Υ r ≤ ℓN ≤ ℓ ( γ − e ζ +1 ≪ ℓ γτ = L τ . (4.19)We can arrange for { Υ r } Rr =1 to be a collection of ( m, I )-buffered subsets of Λ L as follows. It follows from Lemma 3.12 that for any Θ ⊂ Λ L we have P { Θ is L -level spacing for H ω } ≥ − Y µ e − (2 α − L β ( L + 1) d . (4.20)Let F N = N [ r =1 F ( r ) , where F ( r ) = { Φ ⊂ Ξ L,ℓ ; Φ is G -connected and | Φ | = r } . (4.21)Setting F ( r, a ) = { Φ ∈ F r ; a ∈ Φ } for a ∈ Ξ L,ℓ , and noting that each vertex in thegraph G has less than (6 k ℓ − d ≤ (cid:0) ℓ − ς (cid:1) d nearest neighbors , we get |F ( r, a ) | ≤ ( r − (cid:0) ℓ − ς (cid:1) ( r − d = ⇒ |F ( r ) | ≤ ( L + 1) d ( r − (cid:0) ℓ − ς (cid:1) ( r − d = ⇒ |F N | ≤ ( L + 1) d N ! (cid:0) ℓ − ς (cid:1) ( N − d . (4.22)Letting S N denote that the event that the box Λ L and the subsets { Υ Φ } Φ ∈F N areall L -level spacing for H ω , we get from (4.20) and (4.22) that P {S cN } ≤ Y µ (cid:16) L + 1) d N ! (cid:0) ℓ − ς (cid:1) ( N − d (cid:17) ( L + 1) d e − (2 α − L β < e − L ζ (4.23)for sufficiently large L , since ( γ − e ζ < ( γ − β < γβ and ζ < β .We now define the event E N = B N ∩ S N . It follows from (4.9) and (4.23) that P {E N } > − e − L ζ . To finish the proof we need to show that for all ω ∈ E N thebox Λ L is ( M, I ℓ , I )-localizing for H ω , where M is given in (4.8).Let us fix ω ∈ E N . Then we have (4.17), Λ L is level spacing for H ω , and thesubsets { Υ r } Rr =1 constructed in (4.15) are ( m, I )-buffered subsets of Λ L for H ω . Itfollows from (3.108) and Definition 3.6(iii) thatΛ L = ( [ a ∈G Λ Λ L , ℓ − ℓς ℓ ( a ) ) ∪ ( R [ r =1 Υ Λ L , ℓ τ r ) . (4.24)Since ω is fixed, we omit it from the notation. Let { ( ψ λ , λ ) } λ ∈ σ ( H Λ L ) be aneigensystem for H Λ L . Given a ∈ G , let (cid:8) ( ϕ λ ( a ) , λ ( a ) ) (cid:9) λ ( a ) ∈ σ ( H Λ ℓ ( a ) ) be an ( m, I )-localized eigensystem for Λ ℓ ( a ). For r = 1 , , . . . , R , let (cid:8) ( φ ν ( r ) , ν ( r ) ) (cid:9) ν ( r ) ∈ σ ( H Υ r ) be an eigensystem for H Υ r , and set σ Υ r ( H Λ L ) = ne ν ( r ) ; ν ( r ) ∈ σ B ( H Υ r ) o ⊂ σ ( H Λ L ) \ σ G ( H Λ L ) , (4.25)where e ν ( r ) is given in (3.77), which gives σ Υ r ( H Λ L ) ⊂ σ ( H Λ L ) \ σ G Υ r ( H Λ L ), butthe argument actually shows σ Υ r ( H Λ L ) ⊂ σ ( H Λ L ) \ σ G ( H Λ L ). We also set σ B ( H Λ L ) = R [ r =1 σ Υ r ( H Λ L ) ⊂ σ ( H Λ L ) \ σ G ( H Λ L ) . (4.26) We claim σ I ℓ ( H Λ L ) ⊂ σ G ( H Λ L ) ∪ σ B ( H Λ L ) . (4.27)To see this, suppose we have λ ∈ σ I ℓ ( H Λ L ) \ ( σ G ( H Λ L ) ∪ σ B ( H Λ L )). Since Λ L islevel spacing for H , it follows from Lemma 3.5(ii)(c) that | ψ λ ( y ) | ≤ e − m h I ( λ ) ℓ τ for all y ∈ [ a ∈G Λ Λ L , ℓ τ ℓ ( a ) , (4.28)and it follows from Lemma 3.8(ii) that | ψ λ ( y ) | ≤ e − m h Iℓ ( λ ) ℓ τ for all y ∈ R [ r =1 Υ Λ L , ℓ τ r . (4.29)Using λ ∈ I ℓ , (4.24), (4.6), and (3.85) we conclude that (note m ≤ m )1 = k ψ λ k ≤ e − m h Iℓ ( λ ) ℓ τ ( L + 1) d ≤ e − ℓ − ( κ + κ ′ ) ℓ τ ( L + 1) d < , (4.30)a contradiction. This establishes the claim.To finish the proof we need to show that { ( ψ λ , λ ) } λ ∈ σ ( H Λ L ) is an ( M, I ℓ , I )-localized eigensystem for Λ L , where M is given in (4.8). We take λ ∈ σ I ℓ ( H Λ L ), so h I ℓ ( λ ) >
0. In view of (4.27) we consider several cases:(i) Suppose λ ∈ σ G (Λ L ). In this case λ ∈ σ { Λ ℓ ( a λ ) } ( H Λ L ) for some a λ ∈ G .We pick x λ ∈ Λ ( a λ ). In view of (4.24) we consider two cases:(a) If y ∈ Λ Λ L , ℓ − ℓς ℓ ( a ) for some a ∈ G and k y − x λ k ≥ ℓ , we must haveΛ ℓ ( a λ ) ∩ Λ ℓ ( a ) = ∅ , so it follows from (3.61) that λ / ∈ σ { Λ ℓ ( a ) } ( H Λ L ),and, since R ∂ Λ L in Λ ℓ ( a ) y ≥ (cid:4) ℓ − ℓ ς (cid:5) , (3.58) yields | ψ λ ( y ) | ≤ e − m h I ( λ ) ⌊ ℓ − ℓς ⌋ | ψ λ ( y ) | for some y ∈ ∂ Λ L ,ℓ e τ Λ ℓ ( a ) . (4.31)In particular, k y − y k ≤ ℓ − (cid:4) ℓ − ℓ ς (cid:5) ≤ ℓ + ℓ ς + 1 ≤ ℓ +2 ℓ ς . (4.32)(b) If y ∈ Υ Λ L , ℓ τ r for some r ∈ { , , . . . , R } , and k y − x λ k ≥ ℓ +diam Υ r ,we must have Λ ℓ ( a λ ) ∩ Υ r = ∅ . It follows from (3.61) that λ / ∈ σ G Υ r ( H Λ L ), and clearly λ / ∈ σ Υ r ( H Λ L ) in view of (4.25). ThusLemma 3.8(ii) gives | ψ λ ( y ) | ≤ e − m h Iℓ ( λ ) ℓ τ | ψ λ ( y ) | for some y ∈ ∂ Λ L , ℓ τ Υ r . (4.33)In particular, k y − y k ≤ diam Υ r . (4.34)(ii) Suppose λ / ∈ σ G (Λ L ). Then it follows from (4.27) that we must have λ x ∈ σ Υ s ( H Λ L ) for some s ∈ { , , . . . , R } . We pick x λ ∈ Υ Λ L , ℓ τ s . In viewof (4.24) we consider two possibilities:(a) If y ∈ Λ Λ L , ℓ − ℓς ℓ ( a ) for some a ∈ G , and k y − x λ k ≥ ℓ + diam Υ s , wemust have Λ ℓ ( a ) ∩ Υ s = ∅ , and Lemma 3.5(i)(c) yields (4.31).(b) If y ∈ Υ Λ L , ℓ τ r for some r ∈ { , , . . . , R } , and k y − x λ k ≥ diam Υ s +diam Υ r , we must have r = s . Thus Lemma 3.8(ii) yields (4.33). IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 29
Now consider y ∈ Λ L such that k y − x λ k ≥ L τ . Suppose | ψ λ ( y ) | >
0, sinceotherwise there is nothing to prove. We estimate | ψ λ ( y ) | using either (4.31) or(4.33) repeatedly, as appropriate, stopping when we get too close to x λ so we are notin one the cases described above. (Note that this must happen since | ψ x ( y ) | > − m h Iℓ ( λ ) ℓ τ < S times,as long as ℓ +2 ℓ ς S + R X r =1 diam Υ r + 2 ℓ ≤ k y − x λ k . (4.35)In view of (4.19), this can be guaranteed by requiring ℓ +2 ℓ ς S + 5 ℓ ( γ − e ζ +1 + 2 ℓ ≤ k y − x λ k . (4.36)We can thus have S = j ℓ +2 ℓ ς (cid:16) k y − x λ k − ℓ ( γ − e ζ +1 − ℓ (cid:17)k − ≥ ℓ +2 ℓ ς (cid:16) k y − x λ k − ℓ ( γ − e ζ +1 − ℓ (cid:17) − ≥ ℓ +2 ℓ ς (cid:16) k y − x λ k − ℓ ( γ − e ζ +1 − ℓ − ℓ ς (cid:17) ≥ ℓ +2 ℓ ς (cid:16) k y − x λ k − ℓ ( γ − e ζ +1 (cid:17) Thus we conclude that | ψ λ ( y ) | ≤ e − m h I ( λ ) ⌊ ℓ − ℓς ⌋ ℓ +2 ℓ ς (cid:16) k y − x λ k− ℓ ( γ − e ζ +1 (cid:17) ≤ e − Mh I ( λ ) k y − x λ k (4.38)where M ≥ m (cid:16) − C d,m − ℓ − min { − ς,γτ − ( γ − e ζ − } (cid:17) (4.39)= m (cid:16) − C d,m − ℓ − min { γτ − ( γ − e ζ − } (cid:17) ≥ m (cid:16) − C d,m − ℓ − min { κ, − τ ,γτ − ( γ − e ζ − } (cid:17) = m (cid:0) − C d,m − ℓ − ̺ (cid:1) , where we used (1.7), (3.28), and (1.6). In particular, M satisfies (4.8) for sufficientlylarge ℓ .We conclude that { ( ψ λ , λ ) } λ ∈ σ ( H Λ L ) is an ( M, I ℓ , I )-localized eigensystem forΛ L , where M satisfies (4.8), so the box Λ L is ( M, I ℓ , I )-localizing for H ω . (cid:3) Proof of Proposition 4.1.
We assume (4.1) and (4.2) and set L k +1 = L γk , A k +1 = A k (1 − L − κk ), and I k +1 = ( E − A k +1 , E + A k +1 ) for k = 0 , , . . . . Since if a box Λ L is ( M, I ℓ , I )-localizing for H ω it is also ( M, I ℓ )-localizing, if L is sufficiently largeit follows from Lemma 4.2 by an induction argument that for all k = 1 , , . . . wehave (4.3) and (4.4). (cid:3) Proposition 4.3.
Fix m − > . There exists a a finite scale L = L ( d, m − ) withthe following property: Suppose for some scale L ≥ L we have inf x ∈ R d P { Λ L ( x ) is ( m , I ) -localizing for H ω } ≥ − e − L ζ , (4.40) where I = ( E − A , E + A ) ⊂ R , with E ∈ R and A > , and m − L − κ ′ ≤ m ≤ log (cid:0) A d (cid:1) . (4.41) Set L k +1 = L γk , A k +1 = A k (1 − L − κk ) , and I k +1 = ( E − A k +1 , E + A k +1 ) , for k = 0 , , . . . , Then for all k = 1 , , . . . we have inf x ∈ R d P { Λ L ( x ) is ( m k , I k , I k − ) -localizing for H ω } ≥ − e − L ξ for L ∈ [ L k , L k +1 ) , (4.42) where m − L − κ ′ k < m k − (cid:0) − C d,m − L − ̺k − (cid:1) ≤ m k < log (cid:0) A k d (cid:1) , (4.43) with C d,m − as in (4.4) .Proof. We can apply Proposition 4.1, so we have L . Fix L ≥ L , so we have theconclusions of Proposition 4.1.Given a scale L ≥ L , let k = k ( L ) ∈ { , , . . . } be defined by L k ≤ L < L k +1 .We have L k = L γk − ≤ L < L k +1 = L γ k − , so L = L γ ′ k − with γ ≤ γ ′ < γ .We proceed as in Lemma 4.2. We take Λ L = Λ L ( x ), where x ∈ R d , and let C L,L k − = C L,L k − ( x ) be the suitable L k − -cover of Λ L . We let B denote theevent that all boxes in C L,L k − are ( m k − , I k − )-localizing for H ω . It follows from(3.109) and (4.3) that P {B c } ≤ (cid:16) LL ςk − (cid:17) d e − L ζk − = 2 d L ( γ ′ − ς ) dk − e − L ζk − ≤ d L (1 − ςγ ′ ) d e − L ζγ ′ < e − L ξ , (4.44)if L is sufficiently large, since ξγ < ζ . Moreover, letting S denote the event thatthe box Λ L is level spacing for H ω , it follows from Lemma 3.12 that P {S c } ≤ Y µ e − (2 α − L β ( L + 1) d ≤ e − L ξ , (4.45)if L is sufficiently large, since ξ < β . Thus, letting E = B ∩ S , we have P {E } ≥ − e − L ξ . (4.46)It only remains to prove that Λ L is ( m k , I k , I k − )-localizing for H ω for all ω ∈ E .To do so, we fix ω ∈ E and proceed as in the proof of Lemma 4.2. Since ω ∈ B , wehave G = G ( ω ) = Ξ L,L k − . Since ω is now fixed, we omit them from the notation.As in the proof of Lemma 4.2, we get, noticing that ( I k − ) L k − = I k , σ I k ( H Λ L ) ⊂ σ G ( H Λ L ) , (4.47)similarly to (4.27).Let { ( ψ λ , λ ) } λ ∈ σ ( H Λ L ) be an eigensystem for H Λ L . To finish the proof we needto show that the eigensystem is ( m k , I k , I k − )-localized eigensystem for Λ L . Let λ ∈ σ I k ( H Λ L ), then by (4.47) we have we have λ ∈ σ G ( H Λ L ), and hence λ ∈ σ { Λ Lk − ( a λ ) } ( H Λ L ) for some a λ ∈ G . If y ∈ Λ L and k y − x λ k ≥ L k − , it followsfrom (3.108) that y ∈ Λ Λ L , Lk − − Lςk − ℓ ( a ) for some a ∈ G , and moreover Λ L k − ( a λ ) ∩ Λ L k − ( a ) = ∅ , so it follows from (3.61) that λ / ∈ σ { Λ Lk − ( a ) } ( H Λ L ), and, since R ∂ Λ L in Λ Lk − ( a ) y ≥ j L k − − L ςk − k , (3.58) yields | ψ λ ( y ) | ≤ e − m k − , h Ik − ( λ ) (cid:22) Lk − − Lςk − (cid:23) | ψ λ ( y ) | for some y ∈ ∂ Λ L , ( L k − ) e τ Λ L k − ( a ) , (4.48) IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 31 where we need m k − , = m k − , ( L k − ) ≥ m k − (cid:16) − C d,m − L − ( − τ ) k − (cid:17) , (4.49)and we have k y − y k ≤ L k − +2 L ςk − , (4.50)as in (4.32).Now consider y ∈ Λ L such that k y − x λ k ≥ L τ . Suppose | ψ λ ( y ) | >
0, since oth-erwise there is nothing to prove. We estimate | ψ λ ( y ) | using either (4.48) repeatedly,as appropriate, stopping when we get within 2 L k − of x λ . In view of (4.50) , wecan use (4.48) S times, as long as L k − +2 L ςk − S + 2 L k − ≤ k y − x λ k . (4.51)We can thus have S = j L k − +2 L ςk − ( k y − x λ k − L k − ) k − ≥ j L k − +2 L ςk − ( k y − x λ k − L k − ) k − ≥ L k − +2 L ςk − (cid:0) k y − x λ k − L k − − L ςk − (cid:1) ≥ L k − +2 L ςk − ( k y − x λ k − L k − ) . Thus we conclude that | ψ λ ( y ) | ≤ e − m k − , h Ik − ( λ ) (cid:22) Lk − − Lςk − (cid:23) L k − +2 L ςk − ( k y − x λ k− L k − ) (4.53) ≤ e − m k h Ik − ( λ ) k y − x λ k where m k can be taken the same as in (4.4).We conclude that { ( ψ λ , λ ) } λ ∈ σ ( H Λ L ) is an ( m k , I k , I k − )-localized eigensystem forΛ L , where m k satisfies (4.4), so the box Λ L is ( m k , I k , I k − )-localizing for H ω . (cid:3) Proof of Theorem 1.6.
Let L k +1 = L γk , A k +1 = A k (1 − L − κk ), I k +1 = ( E − A k +1 , E + A k +1 ), and m k +1 = m k (cid:0) − C d,m − L − ̺k (cid:1) for k = 0 , , . . . . Given L ≥ L γ = L , let k = k ( L ) ∈ { , , . . . } be defined by L k ≤ L < L k +1 . Since A ∞ (cid:16) − L − κγ (cid:17) − < A k − = ⇒ I L γ ∞ ⊂ I k − , (4.54)we conclude that (1.20) follows from (4.42). (cid:3) Localization
In this section we consider an Anderson model H ω and prove Theorem 1.7 andCorollary 1.8. Lemma 5.1.
Fix m − > , let A > , and I = ( E − A, E + A ) . There exists afinite scale L d,ν,m − such that for all L ≥ L d,ν,m − , a ∈ Z d , letting L = ℓ γ , givenan ( m, I, I ℓ ) -localizing box Λ L ( a ) for the discrete Schr¨odinger operator H , where m satisfies (3.22) , then for all λ ∈ I , max b ∈ Λ L ( a ) W ( a ) λ ( b ) > e − mh Iℓ ( λ ) L = ⇒ min θ ∈ σ LτIℓ ( H Λ L ( a ) ) | λ − θ | < e − L β . (5.1) Proof.
Note that I ⊂ I ℓL ⊂ I ℓ L and inf λ ∈ I h I ℓL ( λ ) ≥ L − κγ . (5.2)Now let λ ∈ I ⊂ I ℓL , and suppose | λ − θ | ≥ e − L β for all θ ∈ σ L τ I ℓ ( H Λ L ( a ) ). Let ψ ∈ V ( λ ). Then it follows from Lemma 3.4(ii) that for large L and b ∈ Λ L ( a ) wehave | ψ ( b ) | ≤ e − m h Iℓ ( λ ) ( L − ) (cid:13)(cid:13) T − a ψ (cid:13)(cid:13) (cid:10) L + 1 (cid:11) ν ≤ e − mh Iℓ ( λ ) L (cid:13)(cid:13) T − a ψ (cid:13)(cid:13) . (5.3) (cid:3) Proof of Theorem 1.7.
Assume Theorem 1.6 holds for some L , and let I = I ∞ , m = m ∞ . Consider L γ ≤ L ∈ N and a ∈ Z d . We haveΛ L ( a ) = [ b ∈ { a + L Z d } , k b − a k≤ L Λ L ( b ) . (5.4)Let Y L,a denote the event that Λ L ( a ) is level spacing for H ω and the boxes Λ L ( b )are ( m, I, I ℓ )-localizing for H ω for all b ∈ (cid:8) a + L Z d (cid:9) with k b − a k ≤ L , where L = ℓ γ . It follows from (1.20) and Lemma 3.12 that P (cid:8) Y cL,a (cid:9) ≤ d e − L ξ + Y µ (5 L + 1) d e − (2 α − L ) β ≤ C µ e − L ξ . (5.5)Suppose ω ∈ Y L,a , λ ∈ I , and max b ∈ Λ L ( a ) W ( a ) ω ,λ ( b ) > e − mh Iℓ ( λ ) L . It followsfrom Lemma 5.1 that min θ ∈ σ LτIℓ ( H Λ L ( a ) ) | λ − θ | < e − L β . Since Λ L ( a ) is levelspacing for H ω , using Lemma 3.5(i)(a) we conclude thatmin θ ∈ σ LτIℓ ( H Λ L ( b ) ) | λ − θ | ≥ e − (5 L ) β − − m h Iℓ ( λ ) L τ − e − L β (5.6) ≥ e − (5 L ) β − − m L − κγ L τ − e − L β ≥ e − L β for all b ∈ (cid:8) a + L Z d (cid:9) with L ≤ k b − a k ≤ L . Since A L ( a ) ⊂ [ b ∈ { a + L Z d } , L ≤k b − a k≤ L Λ L L ( b ) , (5.7)it follows from Lemma 3.4(ii) that for all y ∈ A L ( a ) we have, given ψ ∈ V ω ( λ ), | ψ ( y ) | ≤ e − m h Iℓ ( λ ) ( L − ) (cid:13)(cid:13) T − a ψ (cid:13)(cid:13) h L + 1 i ν ≤ e − mh Iℓ ( λ ) L (cid:13)(cid:13) T − a ψ (cid:13)(cid:13) (5.8) ≤ e − mh Iℓ ( λ ) k y − a k (cid:13)(cid:13) T − a ψ (cid:13)(cid:13) , so we get W ( a ) ω ,λ ( y ) ≤ e − mh Iℓ ( λ ) k y − a k for all y ∈ A L ( a ) . (5.9)Since we have (1.23), we conclude that for ω ∈ Y L,a we always have W ( a ) ω ,λ ( a ) W ( a ) ω ,λ ( y ) ≤ max n e − mh Iℓ ( λ ) k y − a k h y − a i ν , e − mh Iℓ ( λ ) k y − a k o (5.10) ≤ e − mh Iℓ ( λ ) k y − a k for all y ∈ A L ( a ) . (cid:3) IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 33
Proof of Corollary 1.8.
Parts (i) and (ii) are proven in the same way as [GK4,Theorem 7.1(i)-(ii)], using h I L ≥ h I for all L > L ∈ N , setting L k +1 = 2 L k for k = 0 , , , . . . , wehave (recall (1.26)) Z d = Λ L k ( a ) ∪ ∞ [ j = k A L j ( a ) for k = 0 , , , . . . . (5.11)Given k ∈ N , we set L k = 2 k , and consider the event Y k := \ x ∈ Z d ; k x k≤ e d Lξk Y L k ,x , (5.12)where Y L k ,x is the event given in Theorem 1.7. It follows from (1.24) that forsufficiently large k we have P {Y k } ≥ − C (cid:16) d L ξk + 1 (cid:17) d e − L ξk ≥ − d C e − L ξk , (5.13)so we conclude from the Borel-Cantelli Lemma that P {Y ∞ } = 1 , where Y ∞ = lim inf k →∞ Y k . (5.14)We now fix ω ∈ Y ∞ , so there exists k ω ∈ N such that ω ∈ Y L k ,x for all k ω ≤ k ∈ N and x ∈ Z d with k x k ≤ e d L ξk . We set k ′ ω = max { k ω , } . Given x ∈ Z d , wedefine k x ∈ N by e d L ξkx − < k x k ≤ e d L ξkx if k x ≥ , (5.15)and set k x = 1 otherwise. We set k ω ,x = max { k ′ ω , k x } Let x ∈ Z d . If y ∈ B ω ,x = S ∞ k = k ω ,x A L k ( x ), we have y ∈ A L k ( x ) for some k ≥ k ω ,x and ω ∈ Y L k ,x , so it follows from (1.27) that W ( x ) ω ,λ ( x ) W ( x ) ω ,λ ( y ) ≤ e − mh I ( λ ) k y − x k for all λ ∈ I. (5.16)If y / ∈ B ω ,x , we must have k y − x k < L k ω ,x , so for all λ ∈ R , using (1.23) and(5.15), W ( x ) ω ,λ ( x ) W ( x ) ω ,λ ( y ) = W ( x ) ω ,λ ( x ) W ( x ) ω ,λ ( y )e mh I ( λ ) k y − x k e − mh I ( λ ) k y − x k (5.17) ≤ h y − x i ν e mh I ( λ ) k y − x k e − mh I ( λ ) k y − x k ≤ (cid:10) L k ω ,x (cid:11) ν e mh I ( λ ) L k ω ,x e − mh I ( λ ) k y − x k ≤ (cid:28) (cid:16) log k x k d (cid:17) ξ (cid:29) ν e mh I ( λ ) ( log k x k d ) ξ e − mh I ( λ ) k y − x k if k ω ,x = k x (cid:10) L k ′ ω (cid:11) ν e mh I ( λ ) L k ′ ω e − mh I ( λ ) k y − x k if k ω ,x = k ′ ω . Combining (5.16) and (5.17), noting k x k d > e if k x ≥
2, and h I ( λ ) ≤
1, weconclude that for all λ ∈ I with h I ( λ ) > x, y ∈ Z d we have W ( x ) ω ,λ ( x ) W ( x ) ω ,λ ( y ) (5.18) ≤ C m, ω ,ν D (2 d log h x i ) ξ E ν e mh I ( λ )(2 d log h x i ) ξ e − mh I ( λ ) k y − x k ≤ C m, ω ,ν D ( mh I ( λ )) − E ν e ( + ν ) mh I ( λ )(2 d log h x i ) ξ e − mh I ( λ ) k y − x k ≤ C ′ m, ω ,ν ( h I ( λ )) − ν e ( + ν ) mh I ( λ )(2 d log h x i ) ξ e − mh I ( λ ) k y − x k , which is (1.29).Part (iv) follows from (iii), since (1.29) implies | ψ ( x ) | | ψ ( y ) | (5.19) ≤ C m, ω ,ν ( h I ( λ )) − ν (cid:13)(cid:13) T − x ψ (cid:13)(cid:13) e ( + ν ) mh I ( λ )(2 d log h x i ) ξ e − mh I ( λ ) k y − x k ≤ C m, ω ,ν ( h I ( λ )) − ν (cid:13)(cid:13) T − ψ (cid:13)(cid:13) h x i ν e ( + ν ) mh I ( λ )(2 d log h x i ) ξ e − mh I ( λ ) k y − x k , for all x, y ∈ Z d , which is (1.30).Part (v) similarly follows from (iii) using the discrete equivalent of [GK3, Eq. (4.22)]. (cid:3) Connection with the Green’s functions multiscale analysis
Let H ω be an Anderson model. Given Θ ⊂ Z d finite and z / ∈ σ ( H Θ ), we set G Θ ( z ) = ( H Θ − z ) − and G Θ ( z ; x, y ) = (cid:10) δ x , ( H Θ − z ) − δ y (cid:11) for x, y ∈ Θ . (6.1) Definition 6.1.
Let E ∈ R and m > . A box Λ L is said to be ( m, E ) -regular if E / ∈ σ ( H Λ L ) and | G Λ L ( E ; x, y ) | ≤ e − m k x − y k for all x, y ∈ Λ L with k x − y k ≥ L . (6.2)The following theorem is a typical result from the Green’s function multiscaleanalysis. [FroS, FroMSS, DrK, GK1, Kl]. Theorem 6.2.
Let J ⊂ R be a bounded open interval, < ξ < ζ < , and m > .Suppose for some scale L we have inf x ∈ R d P { Λ L ( x ) is ( m, λ ) -regular } ≥ − e − L ζ for all λ ∈ J. (6.3) Then, given m ∈ (0 , m ) , if L is sufficiently large, we have inf x ∈ R d P { Λ L ( x ) is ( m, λ ) -regular } ≥ − e − L ξ for all λ ∈ J, (6.4) and inf x,y ∈ R d k x − y k >L P { for all λ ∈ J either Λ L ( x ) or Λ L ( y ) is ( m, λ ) -regular } ≥ − e − L ξ . (6.5)Here (6.4) are the conclusions of the single energy multiscale analysis, and (6.5)are the conclusions of the energy interval multiscale analysis.Given a bounded open interval J and m >
0, we call a box Λ L ( m, J )-uniformlylocalizing for H if Λ L is level spacing for H , and there exists an eigensystem IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 35 { ( ϕ ν , ν ) } ν ∈ σ ( H Λ L ) for H Λ L such that for all ν ∈ σ J ( H Λ L ) there is x ν ∈ Λ L suchthat ϕ ν is ( x ν , m )-localized. Note that if Λ L is ( m, J )-localizing for H (as in Defi-nition 1.3), it follows from (1.14) that Λ L is ( mr − κ , J r )-uniformly localizing for H for all r > Proposition 6.3.
Let J ⊂ R be a bounded open interval, < ξ ′ < ξ < , and m > . Suppose there exists L such that the Anderson model H ω satisfies (6.5) forall L ≥ L . Then, given m ′ ∈ (0 , m ) , for sufficiently large L we have inf x ∈ R d P { Λ L ( x ) is ( m ′ , J ) -uniformly localizing for H ω } ≥ − e − L ξ ′ . (6.6)Proposition 6.3 is proved exactly as the analogous result in [EK, Proposition 6.4].We now show that the conclusions of Theorem 1.6 imply a result similar to thethe conclusions of Theorem 6.2. Lemma 6.4.
Fix m − > . Let I = ( E − A, E + A ) ⊂ R , with E ∈ R and A > ,and m > . Suppose that Λ L is ( m, I ) -localizing for H , where m − L − κ ′ ≤ m ≤ log (cid:0) A d (cid:1) . (6.7) Then, for sufficiently large L , Λ L is ( m ′′ h I ( λ ) , λ ) -regular for all λ ∈ I L with dist { λ, σ ( H Λ L ) } ≥ e − L β , where m ′′ ≥ m (cid:16) − C d,m − L − (1 − τ ) (cid:17) . (6.8) Proof.
We take E = 0 by replacing the potential V by V − E .Let λ ∈ I with dist { λ, σ ( H Λ L ) } ≥ e − L β . For all t > G Λ L ( λ ) = ( H Λ L − λ ) − = F t,λ ( H Λ L ) + ( H Λ L − λ ) − e − t ( H L − λ ) (6.9)where the function F t,λ ( z ) is defined in (3.6).Let { ( ϕ ν , ν ) } ν ∈ σ ( H Λ L ) be an ( m, I )-localized eigensystem for H Λ L . Let ν ∈ σ I ( H Λ L ) and x, y ∈ Λ L with k x − y k ≥ L . In this case either k x − x ν k ≥ L τ or k y − x ν k ≥ L τ . Say k x − x ν k ≥ L τ , then | ϕ ν ( x ) ϕ ν ( y ) | ≤ ( e − mh I ( ν )( k x − x ν k + k y − x ν k ) ≤ e − mh I ( ν ) k x − y k if k y − x ν k ≥ L τ e − mh I ( ν ) k x − x ν k ≤ e − mh I ( ν )( k x − y k− L τ ) if k y − x ν k < L τ , (6.10)so we conclude that | ϕ ν ( x ) ϕ ν ( y ) | ≤ e − m ′ h I ( ν ) k x − y k , where m ′ ≥ m (1 − L τ − ) . (6.11)Now let P I = χ I ( H Λ L ), ¯ P I = 1 − P I . Since D δ x , ( H Λ L − λ ) − e − t ( H L − λ ) P I δ y E = X µ ∈ σ I ( H Λ L ) ( µ − λ ) − e − t ( µ − λ ) ϕ µ ( x ) ϕ µ ( y ) , (6.12)it follows from (6.11) that (cid:12)(cid:12)(cid:12)D δ x , ( H Λ L − λ ) − e − t ( H L − λ ) P I δ y E(cid:12)(cid:12)(cid:12) ≤ e L β X µ ∈ σ I ( H Λ L ) e − t ( µ − λ ) | ϕ µ ( x ) ϕ µ ( y ) |≤ e L β X µ ∈ σ I ( H Λ L ) e − t ( µ − λ ) e − m ′ h I ( µ ) k x − y k . (6.13) We now take t = m ′ k x − y k A = ⇒ e − t ( µ − λ ) e − m ′ h I ( µ ) k x − y k = e − m ′ h I ( λ ) k x − y k for µ ∈ I, (6.14)obtaining (cid:12)(cid:12)(cid:12)D δ x , ( H Λ L − λ ) − e − t ( H L − λ ) P I δ y E(cid:12)(cid:12)(cid:12) ≤ ( L + 1) d e L β e − m ′ h I ( λ ) k x − y k . (6.15)It follows from Lemma 3.3 that (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) δ x , ( H Λ L − λ ) − e − m ′ k x − y k A ( H L − λ ) ¯ P I δ y (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e L β e − m ′ h I ( λ ) k x − y k , (6.16)so (cid:12)(cid:12)(cid:12)D δ x , ( H Λ L − λ ) − e − t ( H L − λ ) δ y E(cid:12)(cid:12)(cid:12) ≤ L + 1) d e L β e − m ′ h I ( λ ) k x − y k . (6.17)It follows from (3.9), using (6.7), that (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) δ x , F m ′| x − y | A ,λ ( H Λ L ) δ y (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ A − e − m ′ h I ( λ ) | x − y | ≤ m − − L κ ′ e − m ′ h I ( λ ) | x − y | . (6.18)Combining (6.9), (6.17) and (6.18), we get | G Λ L ( λ ; x, y ) | ≤ (cid:16) m − − L κ ′ + 2( L + 1) d e L β (cid:17) e − m ′ h I ( λ ) | x − y | . (6.19)We now require λ ∈ I L , obtaining | G Λ L ( λ ; x, y ) | ≤ e − m ′′ h I ( λ ) k x − y k , (6.20)where m ′′ ≥ m ′ (cid:16) − C d,m − L − (1 − β − κ − κ ′ ) (cid:17) (6.21) ≥ m (cid:16) − C d,m − L − min { − τ, − β − κ − κ ′ } (cid:17) = m (cid:16) − C d,m − L − (1 − τ ) (cid:17) . (cid:3) Proposition 6.5.
Suppose the conclusions of Theorem 1.6 hold for an Andersonmodel H ω , and let I = I ∞ , m = m ∞ . Then, given < ζ ′ < ξ , there exists a finitescale L such that for all L ≥ L we have inf x ∈ R d P { Λ L ( x ) is ( m ′′ h I ( λ ) , λ ) -regular } ≥ − e − L ζ ′ for all λ ∈ I L , (6.22) and inf x,y ∈ R d k x − y k >L P { for λ ∈ I L either Λ L ( x ) or Λ L ( y ) is ( m ′′ h I ( λ ) , λ ) -regular } ≥ − e − L ζ ′ , (6.23) where m ′′ is given in (6.8) . IGENSYSTEM MULTISCALE ANALYSIS IN ENERGY INTERVALS 37
Proof.
Suppose the conclusions of Theorem 1.6 hold for an Anderson model H ω ,and let I = I ∞ , m = m ∞ , and let L ≥ L γ . Since the Wegner estimate gives (seeLemma 3.12 for the notation) P n k G Λ L ( λ ) k ≤ e L β o ≥ − e K α e − αL β ( L + 1) d ≥ − e − L ζ ′ for all λ ∈ R , (6.24)for large L , it follows from (1.20) and Lemma 6.4 that for L large we have (6.22).Now consider two boxes Λ L ( x ) and Λ L ( x ), where x , x ∈ R d , k x − x k > L .Define the events A = { Λ( x ) and Λ( x ) are both ( m, I )-localizing for H ω } , (6.25) B = n dist( σ (Λ L ( x )) , σ (Λ L ( x ))) ≥ − L β o Since k x − x k > L , the boxes are disjoint, so it follows from (1.20) that P {A} ≥ − − L ξ ≥ − e − L ζ ′ , (6.26)and the Wegner estimate between boxes gives P {B} ≥ − e K α e − αL β ( L + 1) d ≥ − e − L ζ ′ , (6.27)so we have P {A ∩ B} ≥ − e − L ζ ′ . (6.28)Moreover, for ω ∈ A ∩ B and λ ∈ R , the boxes Λ( x ) and Λ( x ) are both ( m, I )-localizing, and we must have either (cid:13)(cid:13) G Λ L ( x ) ( λ ) (cid:13)(cid:13) ≤ e L β or (cid:13)(cid:13) G Λ L ( x ) ( λ ) (cid:13)(cid:13) ≤ e L β , sofor λ ∈ I L the previous argument shows that either Λ( x ) or Λ( x ) is ( m ′′ h I ( λ ) , λ )-regular for large L . We proved (6.23). (cid:3) References [A] Aizenman, M.: Localization at weak disorder: some elementary bounds. Rev. Math.Phys. , 1163-1182 (1994)[AG] Aizenman, M., Graf, G. M.: Localization bounds for an electron gas. J. Phys. A ,6783 – 6806 (1998).[ASFH] Aizenman, M., Schenker, J., Friedrich, R., Hundertmark, D.: Finite volume fractional-moment criteria for Anderson localization. Commun. Math. Phys. , 219-253 (2001)[AENSS] Aizenman, M., Elgart, A., Naboko, S., Schenker, J., Stolz, G.: Moment analysis forlocalization in random Schr¨odinger operators. Inv. 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Department of Mathematics; Virginia Tech; Blacksburg, VA, 24061, USA
E-mail address : [email protected] (A. Klein) University of California, Irvine; Department of Mathematics; Irvine, CA92697-3875, USA
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