Eigensystem Bootstrap Multiscale Analysis for the Anderson Model
aa r X i v : . [ m a t h - ph ] N ov Eigensystem Bootstrap Multiscale Analysis for theAnderson Model
Abel Klein ∗ C.S. Sidney Tsang † Abstract
We use a bootstrap argument to enhance the eigensystem multiscale anal-ysis, introduced by Elgart and Klein for proving localization for the Andersonmodel at high disorder. The eigensystem multiscale analysis studies finitevolume eigensystems, not finite volume Green’s functions. It yields purepoint spectrum with exponentially decaying eigenfunctions and dynamicallocalization. The starting hypothesis for the eigensystem bootstrap multi-scale analysis only requires the verification of polynomial decay of the finitevolume eigenfunctions, at some sufficiently large scale, with some minimalprobability independent of the scale. It yields exponential localization offinite volume eigenfunctions in boxes of side L , with the eigenvalues andeigenfunctions labeled by the sites of the box, with probability higher than1 − e − L ξ , for any desired 0 < ξ < Mathematics Subject Classification (2010).
Primary 82B44; Secondary47B80, 60H25, 81Q10.
Keywords.
Anderson localization, Anderson model, eigensystem multiscale anal-ysis
Contents
Introduction 21 Main definitions and results 32 Preliminaries to the multiscale analysis 8 ∗ A.K. was partially supported by the NSF under grant DMS-1301641. † C.S.S.T. was supported by the NSF under grant DMS-1301641.University of California, Irvine; Department of Mathematics; Irvine, CA 92697-3875, USA.E-mail: [email protected], [email protected]. igensystem Bootstrap Multiscale Analysis Introduction
The eigensystem multiscale analysis is a new approach for proving localization forthe Anderson model introduced by Elgart and Klein [EK1]. The usual proofs oflocalization for random Schr¨odinger operators are based on the study of finite vol-ume Green’s functions [FroS, FroMSS, Dr, DrK, S, CoH, FK, GK1, Kl, BoK, GK2,AiM, Ai, AiSFH, AiENSS]. In contrast to the usual strategy, the eigensystem mul-tiscale analysis is based on finite volume eigensystems, not finite volume Green’sfunctions. It treats all energies of the finite volume operator at the same time, es-tablishing level spacing and localization of eigenfunctions in a fixed box with highprobability. A new feature is the labeling of the eigenvalues and eigenfunctions bythe sites of the box.In this paper we use a bootstrap argument as in Germinet and Klein [GK1]to enhance the eigensystem multiscale analysis. It yields exponential localizationof finite volume eigenfunctions in boxes of side L , with the eigenvalues and eigen-functions labeled by the sites of the box, with probability higher than 1 − e − L ξ , forany 0 < ξ <
1. The starting hypothesis for the eigensystem bootstrap multiscaleanalysis only requires the verification of polynomial decay of the finite volumeeigenfunctions, at some sufficiently large scale, with some minimal probability in-dependent of the scale. The advantage of the bootstrap multiscale analysis isthat from the same starting hypothesis we get conclusions that are valid for any0 < ξ < H ε,ω = − ε ∆ + V ω on ℓ ( Z d ) (see Defini-tion 1.1; ε > H ε,ω, Λ , the restrictions of H ε,ω to finite boxes Λ. Theobjects of interest for the eigensystem multiscale analysis are finite volume eigen-systems. An eigensystem { ( ϕ j , λ j ) } j ∈ J for H ε,ω, Λ consists of eigenpairs ( ϕ j , λ j ),where λ j is an eigenvalue for H ε,ω, Λ and ϕ j is a corresponding normalized eigen-function, such that { ϕ j } j ∈ J is an orthonormal basis for the finite dimensionalHilbert space ℓ (Λ). Elgart and Klein [EK1] called a box Λ localizing for H ε,ω if the eigenvalues of H ε,ω, Λ satisfy a level spacing condition, and there exists aneigensystem for H ε,ω, Λ indexed by the sites in the box, { ( ϕ x , λ x ) } x ∈ Λ , with the igensystem Bootstrap Multiscale Analysis { ϕ x } x ∈ Λ exhibiting exponential localization around the label, i.e., | ϕ x ( y ) | ≤ e − m k x − y k for y ∈ Λ distant from x . They showed [EK1, Theorem 1.6]that, fixing ξ ∈ (0 , ε ≪
1) boxes of (sufficiently large) side L are localizing with probability ≥ − e − L ξ , yielding all the usual forms of localiza-tion [EK1, Theorem 1.7 and Corollary 1.8]. More precisely, it is shown in [EK1]that for ξ ∈ (0 ,
1) there exists ε ξ >
0, decreasing as ξ increases, and for ε > L ε , increasing as ε decreases, such that for 0 < ε ≤ ε ξ and L ≥ L ε ξ boxes ofside L are localizing for H ε,ω with probability ≥ − e − L ξ .We use the ideas of Germinet and Klein [GK1] to perform a bootstrap multi-scale analysis for finite volume eigensystems (Theorem 1.6). To start the multiscaleanalysis, we only have to verify a statement of polynomial localization of the eigen-functions with some minimal probability independent of the scale. We concludethat at high disorder boxes of side L are localizing with probability ≥ − e − L ξ for all ξ ∈ (0 , ε >
0, and for each ξ ∈ (0 ,
1) there exists a scale L ε ,ξ , such that for all 0 < ε ≤ ε and L ≥ L ε ,ξ boxes of side L are localizing for H ε,ω with probability ≥ − e − L ξ . How large L needs to be depends on ξ , but the required amount of disorder is independent of ξ .In addition, if we have the conclusions of [EK1, Theorem 1.6] for a fixed ξ ∈ (0 , ξ ′ ∈ (0 ,
1) there exists a scale L ξ ′ , suchthat for all 0 < ε ≤ ε ξ and L ≥ L ξ ′ boxes of side L are localizing for H ε,ω withprobability ≥ − e − L ξ ′ . (Note that ε ξ depends on the fixed ξ but does not dependon ξ ′ .)Recently, Elgart and Klein [EK2] extended the eigensystem multiscale analysisto establish localization for the Anderson model in an energy interval. This exten-sion yields localization at fixed disorder on an interval at the edge of the spectrum(or in the vicinity of a spectral gap), and at a fixed interval at the bottom of thespectrum for sufficiently high disorder. We expect that our bootstrap eigensystemmultiscale analysis can also be extended to energy intervals.Our main definitions and resuts are stated in Section 1. Theorem 1.6 is thebootstrap eigensystem multiscale analysis. Theorem 1.7 gives the high disorderresult for the Anderson model, and yields Theorem 1.8, which encapsulates lo-calization for the Anderson model at high disorder. Theorem 1.6 is proven inSection 4, and Theorem 1.7 is proven in Section 5. In Section 2 we provide nota-tion, definitions and lemmas for the proof of the bootstrap eigensystem multiscaleanalysis. In Section 3 we state the probability estimates for level spacing used inthe proof of the bootstrap eigensystem multiscale analysis. We consider the Anderson model in the following form.
Definition 1.1.
The Anderson model is the random Schr¨odinger operator H ε,ω := − ε ∆ + V ω on ℓ ( Z d ) , (1.1) igensystem Bootstrap Multiscale Analysis ε >
0; ∆ is the (centered) discrete Laplacian:(∆ ϕ )( x ) := X y ∈ Z d , | y − x | =1 ϕ ( y ) for ϕ ∈ ℓ ( Z d ); (1.2) V ω ( x ) = ω x for x ∈ Z d , where ω = { ω x } x ∈ Z d is a family of independent identicallydistributed random variables, with a non-degenerate probability distribution µ with bounded support and H¨older continuous of order α ∈ ( , S µ ( t ) ≤ Kt α for all t ∈ [0 , , (1.3)with S µ ( t ) := sup a ∈ R µ { [ a, a + t ] } the concentration function of the measure µ and K a constant.Given Θ ⊂ Z d , we let T Θ = χ Θ T χ Θ be the restriction of the bounded operator T on ℓ ( Z d ) to ℓ (Θ). If Φ ⊂ Θ ⊂ Z d , we identify ℓ (Φ) with a subset of ℓ (Θ) byextending functions on Φ to functions on Θ that are identically 0 on Θ \ Φ. We write ϕ Φ = χ Φ ϕ if ϕ is a function on Θ. We let k ϕ k = k ϕ k and k ϕ k ∞ = max y ∈ Θ | ϕ ( y ) | for ϕ ∈ ℓ (Θ).For x = ( x , x , . . . , x d ) ∈ R d we set k x k = | x | ∞ = max j =1 , ,...,d | x j | , | x | = | x | = (cid:16)P dj =1 x j (cid:17) , and | x | = P dj =1 | x j | . Given Ξ ⊂ R d , we let diam Ξ =sup x,y ∈ Ξ k y − x k denote its diameter, and set dist( x, Ξ) = inf y ∈ Ξ k y − x k for x ∈ R d .We use boxes in Z d centered at points in 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) . (1.4)We write Λ L to denote a box Λ L ( x ) for some x ∈ R d . We have ( L − d < | Λ L | ≤ ( L + 1) d for L ≥
2, where for a set Θ ⊂ Z d we let | Θ | denote its cardinality.The following definitions are for a fixed discrete Schr¨odinger operator H ε . Weomit ε from the notation (i.e., we write H for H ε , H Θ for H ε, Θ ) when it does notlead to confusion. We always consider scales L ≥ τ ∈ (0 , L ′ = (cid:4) L (cid:5) and L τ = ⌊ L τ ⌋ . (1.5)For fixed q > β, τ ∈ (0 , Definition 1.2.
Let Λ L be a box, x ∈ Λ L , and ϕ ∈ ℓ (Λ L ) with k ϕ k = 1. Then:(i) Given e θ > ϕ is said to be ( x, e θ )-polynomially localized if | ϕ ( y ) | ≤ L − e θ for all y ∈ Λ L with k y − x k ≥ L ′ . (1.6)(ii) Given e s ∈ (0 , ϕ is said to be ( x, e s )-subexponentially localized if | ϕ ( y ) | ≤ e − L e s for all y ∈ Λ L with k y − x k ≥ L ′ . (1.7) igensystem Bootstrap Multiscale Analysis m > ϕ is said to be ( x, m )-localized if | ϕ ( y ) | ≤ e − m k y − x k for all y ∈ Λ L with k y − x k ≥ L τ . (1.8) Definition 1.3.
Let
R >
0, and Θ ⊂ Z d be a finite set such that all eigenvaluesof H Θ are simple (i.e., | σ ( H Θ ) | = | Θ | ). Then:(i) Θ is called R -polynomially level spacing for H Θ if | λ − λ ′ | ≥ R − q for all λ, λ ′ ∈ σ ( H Θ ) , λ = λ ′ .(ii) Θ is called R -level spacing for H Θ if | λ − λ ′ | ≥ e − R β for all λ, λ ′ ∈ σ ( H Θ ) , λ = λ ′ .When Θ = Λ L , a box, and R = L , we will just say that Λ L is polynomially levelspacing for H Λ L , or Λ L is level spacing for H Λ L .Note that R -polynomially level spacing implies R -level spacing for sufficientlylarge R .Given Θ ⊂ Z d , ( ϕ, λ ) is called an eigenpair for H Θ if ϕ ∈ ℓ (Θ), λ ∈ R with k ϕ k = 1, and H Θ ϕ = λϕ (i.e., λ is an eigenvalue for H Θ with a correspondingnormalized eigenfunction ϕ ). A collection { ( ϕ j , λ j ) } j ∈ J of eigenpairs for H Θ iscalled an eigensystem for H Θ if { ϕ j } j ∈ J is an orthonormal basis for ℓ (Θ). We mayrewrite the eigensystem as { ( ψ λ , λ ) } λ ∈ σ ( H Θ ) if all eigenvalues of H Θ are simple. Definition 1.4.
Let Λ L be a box. Then:(i) Given e θ >
0, Λ L will be called e θ -polynomially localizing (PL) for H if thefollowing holds:(a) Λ L is polynomially level spacing for H Λ L .(b) There exists a e θ -polynomially localized eigensystem for H Λ L , that is, aneigensystem { ( ϕ x , λ x ) } x ∈ Λ L for H Λ L such that ϕ x is ( x, e θ )-polynomiallylocalized for all x ∈ Λ L .(ii) Given m ∗ >
0, Λ L will be called m ∗ -mix localizing (ML) for H if the followingholds:(a) Λ L is polynomially level spacing for H Λ L .(b) There exists an m ∗ -localized eigensystem for H Λ L , that is, an eigen-system { ( ϕ x , λ x ) } x ∈ Λ L for H Λ L such that ϕ x is ( x, m ∗ )-localized for all x ∈ Λ L .(iii) Given e s ∈ (0 , L will be called e s -subexponentially localizing (SEL) for H if the following holds:(a) Λ L is level spacing for H Λ L .(b) There exists an e s -subexponentially localized eigensystem for H Λ L , thatis, an eigensystem { ( ϕ x , λ x ) } x ∈ Λ L for H Λ L such that ϕ x is ( x, e s )-subexponentiallylocalized for all x ∈ Λ L . igensystem Bootstrap Multiscale Analysis m >
0, Λ L will be called m -localizing (LOC) for H if the followingholds:(a) Λ L is level spacing for H Λ L .(b) There exists an m -localized eigensystem for H Λ L . Remark 1.5.
It follows immediately from the definition that given e s ∈ (0 , L is m ∗ -mix localizing = ⇒ Λ L is (cid:18) − log m ∗ log L (cid:19) -SEL = ⇒ Λ L is e s -SEL , (1.9)for sufficiently large L . (We consider m ∗ < C a,b,... , C ′ a,b,... , C ( a, b, . . . ), etc., to denote a finite constant depending on the parameters a, b, . . . .Note that C a,b,... may denote different constants in different equations, and evenin the same equation. We will omit the dependence on d and µ from the notation.Given θ > (cid:16) α − + (cid:17) d and 0 < ξ <
1, we introduce the following parameters: • We fix q, p, γ such that d α − < q < (cid:0) θ − d (cid:1) , < p < (2 α − q − d, (1.10)and 1 < γ < min n pp +2 d , θ − d d +4 q o , and note that θ > d + γ (cid:0) d + 2 q (cid:1) > d + 2 q (1.11) • We fix ζ, β, γ, τ such that0 < ξ < ζ < β < γ < < γ < q ζξ , (1.12)and max n γ γ , γβ , ( γ − β +1 γ o < τ < , and note that γ < − τ + γ < τ, and (1.13)0 < ξ < ξγ < ζ < β < τγ < γ < τ < < − βτ − β < γ < τβ . • We fix s such thatmax n γβ, − γ (cid:16) τ − γβ (cid:17)o < s < , (1.14)and note that0 < ζ < β < γβ < s < − τ + − sγ < τ − γβ. (1.15) igensystem Bootstrap Multiscale Analysis • We also let e ζ = ζ + β ∈ ( ζ, β ) , e τ = τ ∈ ( τ,
1) and L e τ = ⌊ L e τ ⌋ . (1.16)In what follows, given θ > (cid:16) α − + (cid:17) d , we fix q, p, γ as in (1.10), and then,given 0 < ξ <
1, we fix ζ, β, γ, τ as in (1.12). We use Definitions 1.2–1.4 withthese fixed q, β, τ , which we omit from the dependence of the constants.
Theorem 1.6.
Let θ > (cid:16) α − + (cid:17) d and ε > . There exists a finite scale L ( ε , θ ) with the following property: Suppose for some ε ∈ (0 , ε ] , L ≥ L ( ε , θ ) ,and ≤ P < d , we have inf x ∈ R d P { Λ L ( x ) is θ -polynomially localizing for H ε,ω } ≥ − P . (1.17) Then, given < ξ < , we can find a finite scale e L = e L ( ε , θ, ξ, L ) and m ξ = m ( ξ, e L ) > such that inf x ∈ R d P { Λ L ( x ) is m ξ -localizing for H ε,ω } ≥ − e − L ξ for all L ≥ e L. (1.18)The eigensystem bootstrap multiscale analysis, stated in Theorem 1.6, is provenin Section 4. It follows from a repeated use of a bootstrap argument, as in [GK1,Section 6], making successive use of Propositions 4.1, 4.3, 4.4, 4.6, 4.8, and 4.9.Propositions 4.1, 4.4, 4.6, and 4.9 are eigensystem multiscale analyses. But thereis a difference in the procedure comparing with the Green’s function bootstrapmultiscale analysis of [GK1]. Unlike the definitions of good boxes for the Green’sfunction multiscale analyses, the definitions of good (i.e., localizing) boxes forthe eigensystem multiscale analyses, given in Definition 1.4, require intermedi-ate scales, namely L and L τ in Definition 1.2. For this reason we only havethe direct implications given in Remark 1.5. Thus the bootstrap between theeigensystem multiscale analyses requires some extra intermediate steps, given inPropositions 4.3 and 4.8.In Section 5 we will prove that we can fulfill the hypotheses of Theorem 1.6,obtaining the following theorem. Theorem 1.7.
There exists ε > such that, given < ξ < , we can find afinite scale e L = e L ( ε , ξ ) and m ξ = m ( ξ, e L ) > such that for all < ε ≤ ε wehave inf x ∈ R d P { Λ L ( x ) is m ξ -localizing for H ε,ω } ≥ − e − L ξ for all L ≥ e L. (1.19)Theorem 1.7 yields all the usual forms of localization. To see this, we introducesome notation and definitions. We fix ν > d , and set h x i = p k x k .A function ψ : Z d → C is called a ν -generalized eigenfunction for H ε if ψ is ageneralized eigenfunction (see (2.12)) and 0 < kh x i − ν ψ k < ∞ . We let V ε ( λ ) denotethe collection of ν -generalized eigenfunctions for H ε with generalized eigenvalue λ ∈ R . igensystem Bootstrap Multiscale Analysis λ ∈ R and a, b ∈ Z d , we set W ( a ) ε,λ ( b ) := ( sup ψ ∈V ε ( λ ) | ψ ( b ) |kh x − a i − ν ψ k if V ε ( λ ) = ∅ . (1.20)Theorem 1.7 yields the following theorem, from which one can derive Ander-son localization (pure point spectrum with exponentially decaying eigenfunctions)dynamical localization, and more, as in [EK1, Corollary 1.8]. Theorem 1.8.
Let H ε,ω be an Anderson model. There exists ε > such that,given ξ ∈ (0 , , we can find a scale b L = b L ( ε , ξ ) and m ξ = m ( ξ, b L ) > , such thatfor all < ε ≤ ε , L ≥ b L with L ∈ N , and a ∈ Z d there exists an event Y ε,L,a with the following properties: (i) Y ε,L,a depends only on the random variables { ω x } x ∈ Λ L ( a ) , and P {Y ε,L,a } ≥ − C ε e − L ξ . (1.21)(ii) For all ω ∈ Y ε,L,a and λ ∈ R we have, with max b ∈ Λ ℓ ( a ) W ( a ) ε,ω,λ ( b ) > e − m ξ L = ⇒ max y ∈ A L ( a ) W ( a ) ε,ω,λ ( y ) ≤ e − m ξ k y − a k , (1.22) where A L ( a ) := (cid:8) y ∈ Z d ; L ≤ k y − a k ≤ L (cid:9) . (1.23) In particular, W ( a ) ε,ω,λ ( a ) W ( a ) ε,ω,λ ( y ) ≤ e − m ξ k y − a k for all y ∈ A L ( a ) . (1.24)Theorem 1.8 is proved in the same way as [EK1, Theorem 1.7]. We consider a fixed discrete Schr¨odinger operator H = − ε ∆ + V on ℓ ( Z d ), where0 < ε ≤ ε for a fixed ε and V is a bounded potential. Let Φ ⊂ Θ ⊂ Z d . We define the boundary, exterior boundary, and interior bound-ary of Φ relative to Θ, respectively, by ∂ Θ Φ = { ( u, v ) ∈ Φ × (Θ \ Φ); | u − v | = 1 } , (2.1) ∂ Θex
Φ = { v ∈ (Θ \ Φ); ( u, v ) ∈ ∂ Θ Φ for some u ∈ Φ } ,∂ Θin
Φ = { u ∈ Φ; ( u, v ) ∈ ∂ Θ Φ for some v ∈ Θ \ Φ } . igensystem Bootstrap Multiscale Analysis H Θ = H Φ ⊕ H Θ \ Φ + ε Γ ∂ Θ Φ on ℓ (Θ) = ℓ (Φ) ⊕ ℓ (Θ \ Φ) , (2.2)where Γ ∂ Θ Φ ( u, v ) = ( − u, v ) or ( v, u ) ∈ ∂ Θ Φ0 otherwise . (2.3)For t ≥ Θ ,t = { y ∈ Φ; Λ t ( y ) ∩ Θ ⊂ Φ } = { y ∈ Φ; dist( y, Θ \ Φ) > ⌊ t ⌋} , (2.4) ∂ Θ ,t in Φ = Φ \ Φ Θ ,t = { y ∈ Φ; dist( y, Θ \ Φ) ≤ ⌊ t ⌋} ,∂ Θ ,t Φ = ∂ Θ ,t in Φ ∪ ∂ Θex Φ . Given a box Λ L ( x ) ⊂ Θ ⊂ Z d we write Λ Θ ,tL ( x ) for (Λ L ( x )) Θ ,t .For a box Λ L ⊂ Θ ⊂ Z d , there exists a unique ˆ v ∈ ∂ Λ L in Θ for each v ∈ ∂ Λ L ex Θsuch that (ˆ v, v ) ∈ ∂ Λ L Θ. Given v ∈ Θ, we define ˆ v as above if v ∈ ∂ Λ L ex Θ, and setˆ v = v otherwise. Note that | ∂ Λ L ex Θ | = | ∂ Λ L Θ | . If L ≥
2, we have | ∂ Θin Λ L | ≤ | ∂ Θex Λ L | = | ∂ Θ Λ L | ≤ s d L d − , where s d = 2 d d. (2.5)To cover a box of side L by boxes of side ℓ < L , we will use suitable covers asin [EK1, Definition 3.10] (also see [GK2, Definition 3.12]). Definition 2.1.
Let Λ L = Λ L ( x ), x ∈ R d be a box in Z d , and let ℓ < L . Asuitable ℓ -cover of Λ L is the collection of boxes C L,ℓ ( x ) = { Λ ℓ ( a ) } a ∈ Ξ L,ℓ , (2.6)where Ξ L,ℓ := { x + ρℓ Z d } ∩ Λ R L with ρ ∈ [ , ] ∩ (cid:8) L − ℓ ℓk ; k ∈ N (cid:9) . (2.7)We call C L,ℓ ( x ) the suitable ℓ -cover of Λ L if ρ = ρ L,ℓ := max (cid:8) [ , ] ∩ (cid:8) L − ℓ ℓk ; k ∈ N (cid:9)(cid:9) .Note that [ , ] ∩ (cid:8) L − ℓ ℓk ; k ∈ N (cid:9) = ∅ if ℓ ≤ L . For a suitable ℓ -cover C L,ℓ ( x ),we have (see [EK1, Lemma 3.11])Λ L = [ a ∈ Ξ L,ℓ Λ Λ L , ℓ ℓ ( a ); (2.8) (cid:0) Lℓ (cid:1) d ≤ L,ℓ = (cid:16) L − ℓρℓ + 1 (cid:17) d ≤ (cid:0) Lℓ (cid:1) d . (2.9) Given Θ ⊂ Z d and an eigensystem { ( ϕ j , λ j ) } j ∈ J for H Θ . We have δ y = X j ∈ J ϕ j ( y ) ϕ j for all y ∈ Θ , (2.10) ψ ( y ) = h δ y , ψ i = X j ∈ J ϕ j ( y ) h ϕ j , ψ i for all ψ ∈ ℓ (Θ) and y ∈ Θ . igensystem Bootstrap Multiscale Analysis ⊂ 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 − ε X y ∈ Θ , | y − x | =1 ψ ( y ) + ( V ( x ) − λ ) ψ ( x ) = 0 for all x ∈ Θ , (2.11)or, equivalently, h ( H Θ − λ ) ϕ, ψ i = 0 for all ϕ ∈ ℓ (Θ) with finite support . (2.12)If ψ ∈ ℓ (Θ), ψ is an eigenfunction for H Θ with eigenvalue λ . We do not requiregeneralized eigenfunctions to be in ℓ (Θ), we only require the pointwise equalityin (2.12). If Θ is finite there is no difference between generalized eigenfunctionsand eigenfunctions. Lemma 2.2.
Consider a box Λ L ⊂ Θ ⊂ Z d , and suppose ( ϕ, λ ) is an eigenpairfor H Λ L . Then: (i) Given e θ > , if ϕ is ( x, e θ ) -polynomially localized for some x ∈ Λ Θ ,L ′ L , wehave dist( λ, σ ( H Θ )) ≤ k ( H Θ − λ ) ϕ k ≤ C d,ε L − ( e θ − d − ) . (2.13)(ii) Given e s ∈ (0 , , if ϕ is ( x, e s ) -subexponentially localized for some x ∈ Λ Θ ,L ′ L ,we have dist( λ, σ ( H Θ )) ≤ k ( H Θ − λ ) ϕ k ≤ e − c L e s , (2.14) where c = c ( L ) ≥ − C d,ε log LL e s . (2.15)(iii) Given m > and τ ∈ (0 , , if ϕ is ( x, m ) localized for some x ∈ Λ Θ ,L τ L , wehave dist( λ, σ ( H Θ )) ≤ k ( H Θ − λ ) ϕ k ≤ e − m L τ , (2.16) where m = m ( L ) ≥ m − C d,ε log LL τ . (2.17) Proof.
We prove part (i), the proofs of (ii) and (iii) are similar. If x ∈ Λ Θ ,L ′ L , wehave dist( x, ∂ Θin Λ L ) ≥ L ′ , thus it follows from [EK1, Lemma 3.2] that k ( H Θ − λ ) ϕ k ≤ ε √ s d L d − k ϕ ∂ Θin Λ L k ∞ ≤ ε √ s d L d − L − e θ (2.18) ≤ ε √ s d L − ( e θ − d − ) . For the following lemmas in this and next subsections, we fix θ > (cid:16) α − + (cid:17) d and 0 < ξ < q, p, γ , ζ, β, γ, τ, s are fixed). Also, when we consider Λ ℓ to be a igensystem Bootstrap Multiscale Analysis ♯ box, where ♯ stands for θ -PL, m ∗ -ML, s -SEL or m -LOC, with m ∗ ≥ m ∗− ( ℓ ) > m ≥ m − ( ℓ ) >
0, we let: L = L ♯ = Y ℓ or ℓ γ if ♯ is θ -PL ℓ γ if ♯ is m ∗ -ML Y ℓ or ℓ γ if ♯ is s -SEL ℓ γ if ♯ is m -LOC and ℓ ♯ = ( ℓ ′ if ♯ is θ -PL or s -SEL ℓ τ if ♯ is m ∗ -ML or m -LOC , (2.19)where Y ≥
1. We will omit the dependence on θ , ξ and Y from the notation.We prove most of the lemmas only for ♯ being θ -PL. The proofs of other casesare similar. Lemma 2.3.
Given Θ ⊂ Z d , let ψ : Θ → C be a generalized eigenfunction for H Θ with generalized eigenvalue λ ∈ R . Consider a ♯ box Λ ℓ ⊂ Θ with a correspondingeigensystem { ( ϕ u , ν u ) } u ∈ Λ ℓ , and suppose for all u ∈ Λ Θ ,ℓ ♯ ℓ we have | λ − ν u | ≥ ( L − q if ♯ is θ -PL or m ∗ -ML e − L β if ♯ is s -SEL or m -LOC . (2.20) Then the following holds for sufficiently large ℓ : (i) Let y ∈ Λ Θ , ℓ ♯ ℓ . Then: (a) If ♯ is θ -PL, we have | ψ ( y ) | ≤ C d,ε L q ℓ − ( θ − d ) | ψ ( y ) | for some y ∈ ∂ Θ , ℓ ′ Λ ℓ . (2.21)(b) If ♯ is s -SEL, we have | ψ ( y ) | ≤ e − c ℓ s | ψ ( y ) | for some y ∈ ∂ Θ , ℓ ′ Λ ℓ , (2.22) where c = c ( ℓ ) ≥ − C d,ε L β ℓ − s . (2.23)(c) If ♯ is m ∗ -ML, we have | ψ ( y ) | ≤ e − m ∗ ℓ τ | ψ ( y ) | for some y ∈ ∂ Θ , ℓ τ Λ ℓ , (2.24) where m ∗ = m ∗ ( ℓ ) ≥ m ∗ − C d,ε γ q log ℓℓ τ . (2.25)(d) If ♯ is m -LOC, we have | ψ ( y ) | ≤ e − m ℓ τ | ψ ( y ) | for some y ∈ ∂ Θ , ℓ τ Λ ℓ , (2.26) where m = m ( ℓ ) ≥ m − C d,ε ℓ γβ − τ . (2.27)(ii) Let y ∈ Λ Θ , ℓ e τ ℓ . Then: (a) If ♯ is m ∗ -ML, we have | ψ ( y ) | ≤ e − m ∗ k y − y k | ψ ( y ) | for some y ∈ ∂ Θ ,ℓ e τ Λ ℓ , (2.28) where m ∗ = m ∗ ( ℓ ) ≥ m ∗ (cid:16) − ℓ τ − (cid:17) − C d,ε γ q log ℓℓ e τ . (2.29) igensystem Bootstrap Multiscale Analysis If ♯ is m -LOC, we have | ψ ( y ) | ≤ e − m k y − y k | ψ ( y ) | for some y ∈ ∂ Θ ,ℓ e τ Λ ℓ , (2.30) where m = m ( ℓ ) ≥ m (cid:16) − ℓ τ − (cid:17) − C d,ε ℓ γβ − e τ . (2.31) Proof.
Let y ∈ Λ ℓ , we have (see (2.10)) ψ ( y ) = X u ∈ Λ ℓ ϕ u ( y ) h ϕ u , ψ i = X u ∈ Λ Θ ,ℓ ′ ℓ ϕ u ( y ) h ϕ u , ψ i + X u ∈ ∂ Θ ,ℓ ′ in Λ ℓ ϕ u ( y ) h ϕ u , ψ i . (2.32)If u ∈ Λ Θ ,ℓ ′ ℓ , we have | λ − ν u | ≥ L − q by (2.20). Using (2.12), we get h ϕ u , ψ i = ( λ − ν u ) − h ϕ u , ( H Θ − ν u ) ψ i = ( λ − ν u ) − h ( H Θ − ν u ) ϕ u , ψ i . (2.33)It follows from [EK1, Lemma 3.2] that | ϕ u ( y ) h ϕ u , ψ i| ≤ L q ε X v ∈ ∂ Θex Λ ℓ | ϕ u ( y ) ϕ u (ˆ v ) || ψ ( v ) | . (2.34)If v ′ ∈ ∂ Θin Λ ℓ , we have k v ′ − u k ≥ ℓ ′ , so (1.6) gives | ϕ u ( v ′ ) | ≤ ℓ − θ . It follows from(2.34) and k ϕ u k = 1 that | ϕ u ( y ) h ϕ u , ψ i| ≤ εL q ℓ − θ X v ∈ ∂ Θex Λ ℓ | ψ ( v ) | ≤ εs d L q ℓ − ( θ − d +1) | ψ ( v ) | (2.35)for some v ∈ ∂ Θex Λ ℓ . Therefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X u ∈ Λ Θ ,ℓ ′ ℓ ϕ u ( y ) h ϕ u , ψ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ εs d L q ℓ − ( θ − d +1) | ψ ( v ) | (2.36)for some v ∈ ∂ Θex Λ ℓ .Let y ∈ Λ Θ , ℓ ′ ℓ . If u ∈ ∂ Θ ,ℓ ′ in Λ ℓ , we have k u − y k ≥ ℓ ′ − ℓ ′ = ℓ ′ , thus (1.6) gives | ϕ u ( y ) | ≤ ℓ − θ , and hence (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X u ∈ ∂ Θ ,ℓ ′ in Λ ℓ ϕ u ( y ) h ϕ u , ψ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ℓ − ( θ − d ) k ψχ Λ ℓ k ≤ ℓ − ( θ − d ) | ψ ( v ) | (2.37)for some v ∈ Λ ℓ . Combining (2.32), (2.36) and (2.37), we conclude that | ψ ( y ) | ≤ (1 + 2 ε s d ) L q ℓ − ( θ − d ) | ψ ( y ) | (2.38)for some y ∈ Λ ℓ ∪ ∂ Θex Λ ℓ . If y ∂ Θ , ℓ ′ Λ ℓ we repeat the procedure to estimate | ψ ( y ) | . Since we can suppose ψ ( y ) = 0 without loss of generality, the proceduremust stop after finitely many times, and at that time we must have (2.21). igensystem Bootstrap Multiscale Analysis ♯ being m ∗ -ML. The proof for ♯ being m -LOC issimilar. Let y ∈ Λ Θ ,ℓ e τ ℓ , then k y − v ′ k ≥ ℓ e τ for v ′ ∈ ∂ Θin Λ ℓ . Thus for u ∈ Λ Θ ,ℓ τ ℓ and v ′ ∈ ∂ Θin Λ ℓ we have | ϕ u ( y ) ϕ u ( v ′ ) | ≤ ( e − m ∗ ( k y − u k + k v ′ − u k ) ≤ e − m ∗ k v ′ − y k if k y − u k ≥ ℓ τ e − m ∗ k v ′ − u k ≤ e − m ′ k v ′ − y k if k y − u k < ℓ τ , (2.39)where m ′ ≥ m ∗ (cid:16) − ℓ τ − e τ (cid:17) = m ∗ (cid:16) − ℓ τ − (cid:17) , (2.40)since for k y − u k < ℓ τ , we have k v ′ − u k ≥ k v ′ − y k − k y − u k ≥ k v ′ − y k − ℓ τ ≥ k v ′ − y k (cid:16) − ℓ τ ℓ e τ (cid:17) . (2.41)Combining (2.34) and (2.39), we conclude that | ϕ u ( y ) h ϕ u , ψ i| ≤ εL q X v ∈ ∂ Θex Λ ℓ e − m ′ ( k v − y k− | ψ ( v ) | (2.42) ≤ εs d ℓ γ q + d − e − m ′ ( k v − y k− | ψ ( v ) | ≤ e − m ′ k v − y k | ψ ( v ) | for some v ∈ ∂ Θex Λ ℓ , where we used k v − y k ≥ ℓ e τ and took m ′ ≥ m ′ (cid:16) − ℓ e τ (cid:17) − C d,ε γ q log ℓℓ e τ ≥ m ∗ (cid:16) − ℓ τ − (cid:17) − C d,ε γ q log ℓℓ e τ . (2.43)Therefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X u ∈ Λ Θ ,ℓτℓ ϕ u ( y ) h ϕ u , ψ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ℓ d e − m ′ k v − y k | ψ ( v ) | ≤ e − m ′ k v − y k | ψ ( v ) | (2.44)for some v ∈ ∂ Θex Λ ℓ , where m ′ ≥ m ′ − C d log ℓℓ e τ ≥ m ∗ (cid:16) − ℓ τ − (cid:17) − C d,ε γ q log ℓℓ e τ . (2.45)If u ∈ ∂ Θ ,ℓ τ in Λ ℓ we have k u − y k ≥ ℓ e τ − ℓ τ > ℓ e τ , thus (1.8) gives | ϕ u ( y ) | ≤ e − m ∗ k u − y k . Also, (1.8) implies | ϕ u ( v ) | ≤ e m ∗ ℓ τ e − m ∗ k v − u k for all v ∈ Λ ℓ . (2.46)Therefore |h ϕ u , ψ i| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X v ∈ Λ ℓ ϕ u ( v ) ψ ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X v ∈ Λ ℓ e − m ∗ ( k v − u k− ℓ τ ) | ψ ( v ) | , (2.47) igensystem Bootstrap Multiscale Analysis | ϕ u ( y ) h ϕ u , ψ i| ≤ X v ∈ Λ ℓ e − m ∗ ( k u − y k− ℓ τ + k v − u k ) | ψ ( v ) | (2.48) ≤ ( ℓ + 1) d e − m ∗ ( k u − y k− ℓ τ ) − m ∗ k v − u k | ψ ( v ) |≤ e − m ′ k u − y k− m ∗ k v − u k | ψ ( v ) |≤ e − m ′ max {k v − y k , k u − y k} | ψ ( v ) | ≤ e − m ′ max {k v − y k , ℓ e τ } | ψ ( v ) | for some v ∈ Λ ℓ , where we used k u − y k ≥ ℓ e τ and took m ′ ≥ m ∗ (cid:16) − ℓ τ − (cid:17) − C d log ℓℓ e τ . (2.49)Therefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X u ∈ ∂ Θ ,ℓτ in Λ ℓ ϕ u ( y ) h ϕ u , ψ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ℓ d e − m ′ max {k v − y k , ℓ e τ } | ψ ( v ) | (2.50) ≤ e − m ′ max {k v − y k , ℓ e τ } | ψ ( v ) | for some v ∈ Λ ℓ , where m ′ ≥ m ∗ ′ − C d log ℓℓ e τ ≥ m ∗ (cid:16) − ℓ τ − (cid:17) − C d log ℓℓ e τ . (2.51)Combining (2.32), (2.44), and (2.50), we conclude that | ψ ( y ) | ≤ e − m ∗ max {k y − y k , ℓ e τ } | ψ ( y ) | for some y ∈ Λ ℓ ∪ ∂ Θex Λ ℓ , (2.52)where m ∗ is given in (2.29). If y ∂ Θ ,ℓ e τ Λ ℓ we repeat the procedure to estimate | ψ ( y ) | . Since we can suppose ψ ( y ) = 0 without loss of generality, the proceduremust stop after finitely many times, and at that time we must have | ψ ( y ) | ≤ e − m ∗ max {k e y − y k , ℓ e τ } | ψ ( e y ) | for some e y ∈ ∂ Θ ,ℓ e τ Λ ℓ . (2.53)If y ∈ Λ Θ , ℓ e τ ℓ , (2.28) follows immediately from (2.53). Lemma 2.4.
Given a finite set Θ ⊂ Z d , let { ( ψ λ , λ ) } λ ∈ σ ( H Θ ) be an eigensystemfor H Θ .Then the following holds for sufficiently large ℓ : (i) Let Λ ℓ ( a ) ⊂ Θ , where a ∈ R d , be a ♯ -localizing box with a correspondingeigensystem n ( ϕ ( a ) x , λ ( a ) x ) o x ∈ Λ ℓ ( a ) , and let Θ be L -polynomially level spacingfor H if ♯ is θ -PL or m ∗ -ML, L -level spacing for H if ♯ is s -SEL or m -LOC. (a) There exists an injection x ∈ Λ Θ ,ℓ ♯ ℓ ( a ) e λ ( a ) x ∈ σ ( H Θ ) , (2.54) such that for all x ∈ Λ Θ ,ℓ ♯ ℓ ( a ) : igensystem Bootstrap Multiscale Analysis i. If ♯ is θ -PL, we have (cid:12)(cid:12)(cid:12)e λ ( a ) x − λ ( a ) x (cid:12)(cid:12)(cid:12) ≤ C d,ε ℓ − ( θ − d − ) , (2.55) and, multiplying each ϕ ( a ) x by a suitable phase factor, (cid:13)(cid:13)(cid:13) ψ e λ ( a ) x − ϕ ( a ) x (cid:13)(cid:13)(cid:13) ≤ C d,ε L q ℓ − ( θ − d − ) . (2.56) ii. If ♯ is s -SEL, we have (cid:12)(cid:12)(cid:12)e λ ( a ) x − λ ( a ) x (cid:12)(cid:12)(cid:12) ≤ e − c ℓ s , with c = c ( ℓ ) as in (2.15) , (2.57) and, multiplying each ϕ ( a ) x by a suitable phase factor, (cid:13)(cid:13)(cid:13) ψ e λ ( a ) x − ϕ ( a ) x (cid:13)(cid:13)(cid:13) ≤ − c ℓ s e L β . (2.58) iii. If ♯ is m ∗ -ML, we have (cid:12)(cid:12)(cid:12)e λ ( a ) x − λ ( a ) x (cid:12)(cid:12)(cid:12) ≤ e − m ∗ ℓ τ , with m ∗ = m ∗ ( ℓ ) as in (2.17) , (2.59) and, multiplying each ϕ ( a ) x by a suitable phase factor, (cid:13)(cid:13)(cid:13) ψ e λ ( a ) x − ϕ ( a ) x (cid:13)(cid:13)(cid:13) ≤ − m ∗ ℓ τ L q . (2.60) iv. If ♯ is m -LOC, we have (cid:12)(cid:12)(cid:12)e λ ( a ) x − λ ( a ) x (cid:12)(cid:12)(cid:12) ≤ e − m ℓ τ , with m = m ( ℓ ) as in (2.17) , (2.61) and, multiplying each ϕ ( a ) x by a suitable phase factor, (cid:13)(cid:13)(cid:13) ψ e λ ( a ) x − ϕ ( a ) x (cid:13)(cid:13)(cid:13) ≤ − m ℓ τ e L β . (2.62)(b) Set σ { a } ( H Θ ) := ne λ ( a ) x ; x ∈ Λ Θ ,ℓ ♯ ℓ ( a ) o . (2.63) Then if λ ∈ σ { a } ( H Θ ) , for all y ∈ Θ \ Λ ℓ ( a ) we have | ψ λ ( y ) | ≤ C d,ε L q ℓ − ( θ − d − ) if ♯ is θ -PL − c ℓ s e L β if ♯ is s -SEL − m ∗ ℓ τ L q if ♯ is m ∗ -ML − m ℓ τ e L β if ♯ is m -LOC . (2.64) igensystem Bootstrap Multiscale Analysis If λ ∈ σ ( H Θ ) \ σ { a } ( H Θ ) , for all x ∈ Λ Θ ,ℓ ♯ ℓ ( a ) we have (cid:12)(cid:12)(cid:12) λ − λ ( a ) x (cid:12)(cid:12)(cid:12) ≥ ( L − q if ♯ is θ -PL or m ∗ -ML e − L β if ♯ is s -SEL or m -LOC , (2.65) and for all y ∈ Λ Θ , ℓ ♯ ℓ ( a ) , | ψ λ ( y ) | ≤ C d,ε L q ℓ − ( θ − d ) | ψ λ ( y ) | if ♯ is θ -PL e − c ℓ s | ψ λ ( y ) | if ♯ is s -SEL e − m ∗ ℓ τ | ψ λ ( y ) | if ♯ is m ∗ -ML e − m ℓ τ | ψ λ ( y ) | if ♯ is m -LOC (2.66) for some y ∈ ∂ Θ , ℓ ♯ Λ ℓ ( a ) , where c = c ( ℓ ) as in (2.23) , m ∗ = m ∗ ( ℓ ) as in (2.25) , m = m ( ℓ ) as in (2.27) . Moreover, for all y ∈ Λ Θ , ℓ e τ ℓ ( a ) , | ψ λ ( y ) | ≤ ( e − m ∗ k y − y k | ψ λ ( y ) | if ♯ is m ∗ -ML e − m k y − y k | ψ λ ( y ) | if ♯ is m -LOC (2.67) for some y ∈ ∂ Θ ,ℓ e τ Λ ℓ ( a ) , where m ∗ = m ∗ ( ℓ ) as in (2.29) , m = m ( ℓ ) as in (2.31) . (ii) Let { Λ ℓ ( a ) } a ∈G , where G ⊂ R d such that Λ ℓ ( a ) ⊂ Θ for all a ∈ G , be acollection of ♯ boxes with corresponding eigensystems n ( ϕ ( a ) x , λ ( a ) x ) o x ∈ Λ ℓ ( a ) and let Θ be L -polynomially level spacing for H if ♯ is θ -PL or m ∗ -ML, L -level spacing for H if ♯ is s -SEL or m -LOC. Set E Θ G ( λ ) = n λ ( a ) x ; a ∈ G , x ∈ Λ Θ ,ℓ ♯ ℓ ( a ) , e λ ( a ) x = λ o for λ ∈ σ ( H Θ ) , (2.68) σ G ( H Θ ) = (cid:8) λ ∈ σ ( H Θ ); E Θ G ( λ ) = ∅ (cid:9) = [ ; a ∈G σ { a } ( H Θ ) . (a) For a, b ∈ G , a = b , if x ∈ Λ Θ ,ℓ ♯ ℓ ( a ) and y ∈ Λ Θ ,ℓ ♯ ℓ ( b ) , λ ( a ) x , λ ( b ) x ∈ E Θ G ( λ ) = ⇒ k x − y k < ℓ ♯ . (2.69) As a consequence, Λ ℓ ( a ) ∩ Λ ℓ ( b ) = ∅ = ⇒ σ { a } ( H Θ ) ∩ σ { b } ( H Θ ) = ∅ . (2.70)(b) If λ ∈ σ G ( H Θ ) , we have for all y ∈ Θ \ Θ G , where Θ G := S a ∈G Λ ℓ ( a ) , | ψ λ ( y ) | ≤ C d,ε L q ℓ − ( θ − d − ) if ♯ is θ -PL − c ℓ s e L β if ♯ is s -SEL − m ∗ ℓ τ L q if ♯ is m ∗ -ML − m ℓ τ e L β if ♯ is m -LOC . (2.71) igensystem Bootstrap Multiscale Analysis If λ ∈ σ ( H Θ ) \ σ G ( H Θ ) , we have for all y ∈ Θ ′G := S a ∈G Λ Θ , ℓ ♯ ℓ ( a ) , | ψ λ ( y ) | ≤ C d,ε L q ℓ − ( θ − d ) if ♯ is θ -PL e − c ℓ s if ♯ is s -SEL e − m ∗ ℓ τ if ♯ is m ∗ -ML e − m ℓ τ if ♯ is m -LOC . (2.72)(d) If | Θ | ≤ ( L + 1) d , we have | Θ ′G | ≤ | σ G ( H Θ ) | ≤ | Θ G | . (2.73) Proof.
Let Λ ℓ ( a ) ⊂ Θ, where a ∈ R d , be a θ -polynomially localizing box with acorresponding eigensystem n ( ϕ ( a ) x , λ ( a ) x ) o x ∈ Λ ℓ ( a ) . It follows from Lemma 2.2 thatthere exists e λ ( a ) x ∈ σ ( H Θ ) satisfying (2.55) for x ∈ Λ Θ ,ℓ ′ ℓ ( a ). e λ ( a ) x is unique sinceΘ is L -polynomially level spacing for H Θ and q < γ q < θ − d − . Moreover, wehave e λ ( a ) x = e λ ( a ) y if x, y ∈ Λ Θ ,ℓ ′ ℓ ( a ), x = y , since (cid:12)(cid:12)(cid:12)e λ ( a ) x − e λ ( a ) y (cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) λ ( a ) x − λ ( a ) y (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)e λ ( a ) x − λ ( a ) x (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)e λ ( a ) y − λ ( a ) y (cid:12)(cid:12)(cid:12) (2.74) ≥ ℓ − q − C d,ε ℓ − ( θ − d − ) ≥ ℓ − q , Λ ℓ ( a ) is polynomially level spacing for H Λ ℓ ( a ) , and q < θ − d − . (2.56) followsfrom [EK1, Lemma 3.3].If λ ∈ σ { a } ( H Θ ), we have λ = e λ ( a ) x for some x ∈ Λ Θ ,ℓ ′ ℓ ( a ), thus (2.64) followsfrom (2.56) as ϕ ( a ) x ( y ) = 0 for all y ∈ Θ \ Λ ℓ ( a ).If λ ∈ σ ( H Θ ) \ σ { a } ( H Θ ), for all x ∈ Λ Θ ,ℓ ′ ℓ ( a ) we have (cid:12)(cid:12)(cid:12) λ − λ ( a ) x (cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) λ − e λ ( a ) x (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)e λ ( a ) x − λ ( a ) x (cid:12)(cid:12)(cid:12) ≥ L − q − C d,ε ℓ − ( θ − d − ) ≥ L − q , (2.75)since Θ is L -polynomially level spacing for H Θ , we have (2.55), and q < γ q <θ − d − . Therefore (2.66) follows from Lemma 2.3(i). (Note that (2.67) followsfrom Lemma 2.3(ii).)Now let { Λ ℓ ( a ) } a ∈G , where G ⊂ R d such that Λ ℓ ( a ) ⊂ Θ for all a ∈ G , bea collection of θ -polynomially localizing boxes with corresponding eigensystems n ( ϕ ( a ) x , λ ( a ) x ) o x ∈ Λ ℓ ( a ) . Let λ ∈ σ ( H Θ ), a, b ∈ G , a = b , x ∈ Λ Θ ,ℓ ′ ℓ ( a ) and y ∈ Λ Θ ,ℓ ′ ℓ ( b ). Assume λ ( a ) x , λ ( b ) x ∈ E Θ G ( λ ), then it follows from (2.56) that (cid:13)(cid:13)(cid:13) ϕ ( a ) x − ϕ ( b ) y (cid:13)(cid:13)(cid:13) ≤ C d,ε L q ℓ − ( θ − d − ) , (2.76)thus (cid:12)(cid:12)(cid:12)D ϕ ( a ) x , ϕ ( b ) y E(cid:12)(cid:12)(cid:12) ≥ ℜ D ϕ ( a ) x , ϕ ( b ) y E ≥ − C d,ε L q ℓ − ( θ − d − ) . (2.77) igensystem Bootstrap Multiscale Analysis k x − y k ≥ ℓ ′ = ⇒ (cid:12)(cid:12)(cid:12)D ϕ ( a ) x , ϕ ( b ) y E(cid:12)(cid:12)(cid:12) ≤ ( ℓ + 1) d ℓ − θ . (2.78)Combining (2.77) and (2.78), we conclude that λ ( a ) x , λ ( b ) x ∈ E Θ G ( λ ) = ⇒ k x − y k < ℓ ′ . (2.79)To prove (2.70), let a, b ∈ G , a = b . Assume Λ ℓ ( a ) ∩ Λ ℓ ( b ) = ∅ , then x ∈ Λ Θ ,ℓ ′ ℓ ( a ) and y ∈ Λ Θ ,ℓ ′ ℓ ( b ) = ⇒ k x − y k ≥ ℓ ′ , (2.80)thus it follows from (2.69) that σ { a } ( H Θ ) ∩ σ { b } ( H Θ ) = ∅ .Parts (ii)(b) and (ii)(c) follow immediately from parts (i)(b) and (i)(c) respec-tively. To prove part (ii)(d), we let P G be the orthogonal projection onto the spanof { ψ λ ; λ ∈ σ G ( H Θ ) } . (2.72) gives k (1 − P G ) δ y k ≤ C d,ε L q ℓ − ( θ − d ) | Θ | for all y ∈ Θ ′G , (2.81)thus k (1 − P G ) χ Θ ′G k ≤ | Θ ′G | | Θ | C d,ε L q ℓ − ( θ − d ) ≤ | Θ | C d,ε L q ℓ − ( θ − d ) . (2.82)If | Θ | ≤ ( L + 1) d , we have k (1 − P G ) χ Θ ′G k ≤ ( L + 1) d C d,ε L q ℓ − ( θ − d ) < d + q < γ ( d + q ) < θ − d , so it follows from [EK1, Lemma A.1] that | Θ ′G | = tr χ Θ ′G ≤ tr P G = | σ G ( H Θ ) | . (2.84)Using a similar argument and (2.71), we can prove | σ G ( H Θ ) | ≤ | Θ G | . For boxes Λ ℓ ⊂ Λ L that are not ♯ for H , we will surround them with a buffer of ♯ boxes and study eigensystems for the augmented subset. Definition 2.5.
Let Λ L = Λ L ( x ) and x ∈ R d . Υ ⊂ Λ L is called a ♯ -bufferedsubset of Λ L , where ♯ stands for θ -PL, s -SEL, m ∗ -ML or m -LOC, if the followingholds:(i) Υ is a connected set in Z d of the formΥ = J [ j =1 Λ R j ( a j ) ∩ Λ L , (2.85)where J ∈ N , a , a , . . . , a J ∈ Λ R L , and ℓ ≤ R j ≤ L for j = 1 , , . . . , J . igensystem Bootstrap Multiscale Analysis L -polynomially level spacing for H if ♯ is θ -PL or m ∗ -ML, L -level spacingfor H if ♯ is s -SEL or m -LOC.(iii) There exists G Υ ⊂ Λ R L such that:(a) For all a ∈ G Υ we have Λ ℓ ( a ) ⊂ Υ, Λ ℓ ( a ) is a ♯ box for H .(b) For all y ∈ ∂ Λ L in Υ there exists a y ∈ G Υ such that y ∈ Λ Υ , ℓ ♯ ℓ ( a y ).In this case we set b Υ = [ a ∈G Υ Λ ℓ ( a ) , c Υ ′ = [ a ∈G Υ Λ Υ , ℓ ♯ ℓ ( a ) , b Υ = Υ \ b Υ , and c Υ ′ = Υ \ c Υ ′ . (2.86)( b Υ = Υ G Υ and c Υ ′ = Υ ′G Υ in the notation of Lemma 2.4.) Lemma 2.6.
Given a ♯ -buffered subset Υ of Λ L , let { ( ψ ν , ν ) } ν ∈ σ ( H Υ ) be an eigen-system for H Υ . Let G = G Υ and set σ B ( H Υ ) = σ ( H Υ ) \ σ G ( H Υ ) , (2.87) where σ G ( H Υ ) is as in (2.68) . Then the following holds for sufficiently large ℓ : (i) If ν ∈ σ B ( H Υ ) we have for all y ∈ c Υ ′ : | ψ λ ( y ) | ≤ C d,ε L q ℓ − ( θ − d ) if ♯ is θ -PL e − c ℓ s , with c = c ( ℓ ) as in (2.23) if ♯ is s -SEL e − m ∗ ℓ τ , with m ∗ = m ∗ ( ℓ ) as in (2.25) if ♯ is m ∗ -ML e − m ℓ τ , with m = m ( ℓ ) as in (2.27) if ♯ is m -LOC , (2.88) and (cid:12)(cid:12)(cid:12) b Υ (cid:12)(cid:12)(cid:12) ≤ | σ B ( H Υ ) | ≤ (cid:12)(cid:12)(cid:12)c Υ ′ (cid:12)(cid:12)(cid:12) . (2.89)(ii) Let Λ L be polynomially level spacing for H if ♯ is θ -PL or m ∗ -ML, levelspacing for H if ♯ is s -SEL or m -LOC, and let { ( φ λ , λ ) } λ ∈ σ ( H Λ L ) be aneigensystem for H Λ L . There exists an injection ν ∈ σ B ( H Υ ) e ν ∈ σ ( H Λ L ) \ σ G ( H Λ L ) , (2.90) such that for all ν ∈ σ B ( H Υ ) : (a) If ♯ is θ -PL, we have | e ν − ν | ≤ C d,ε L d + q ℓ − ( θ − d ) , (2.91) and, multiplying each ψ ν by a suitable phase factor, k φ e ν − ψ ν k ≤ C d,ε L d +2 q ℓ − ( θ − d ) . (2.92) igensystem Bootstrap Multiscale Analysis If ♯ is s -SEL, we have | e ν − ν | ≤ e − c ℓ s , where c = c ( ℓ ) ≥ − C d,ε L β ℓ − s , (2.93) and, multiplying each ψ ν by a suitable phase factor, k φ e ν − ψ ν k ≤ − c ℓ s e L β . (2.94)(c) If ♯ is m ∗ -ML, we have | e ν − ν | ≤ e − m ∗ ℓ τ , where m ∗ = m ∗ ( ℓ ) ≥ m ∗ − C d,ε γ q log ℓℓ τ , (2.95) and, multiplying each ψ ν by a suitable phase factor, k φ e ν − ψ ν k ≤ − m ∗ ℓ τ L q . (2.96)(d) If ♯ is m -LOC, we have | e ν − ν | ≤ e − m ℓ τ , where m = m ( ℓ ) ≥ m − C d,ε ℓ γβ − τ , (2.97) and, multiplying each ψ ν by a suitable phase factor, k φ e ν − ψ ν k ≤ − m ℓ τ e L β . (2.98) Proof.
Part (i) follows immediately from Lemma 2.4(ii)(c) and (ii)(d).Let Λ L be polynomially level spacing, and let { ( φ λ , λ ) } λ ∈ σ ( H Λ L ) be an eigen-system for H Λ L . It follows from [EK1, Lemma 3.2] that for ν ∈ σ B ( H Υ ) we have k ( H Λ L − ν ) ψ ν k ≤ (2 d − ε | ∂ Λ L ex Υ | (cid:13)(cid:13)(cid:13) ϕ ∂ Λ L in Υ (cid:13)(cid:13)(cid:13) ∞ ≤ (2 d − εL d C d,ε L q ℓ − ( θ − d ) (2.99) ≤ C d,ε L d + q ℓ − ( θ − d ) , where we used ∂ Λ L in Υ ⊂ c Υ ′ and (2.88). The map in (2.90) is a well defined injectioninto σ ( H Λ L ) since Λ L and Υ are L -polynomially level spacing for H , and (2.92)follows from (2.91) and [EK1, Lemma 3.3].To show e ν σ G ( H Λ L ) for all ν ∈ σ B ( H Υ ), we assume e ν ∈ σ G ( H Λ L ) for some ν ∈ σ B ( H Υ ). Then there is a ∈ G and x ∈ Λ Λ L ,ℓ ′ ℓ ( a ) such that λ ( a ) x ∈ E Λ L G ( e ν ).On the other hand, λ ( a ) x ∈ E Υ G ( λ ) for some λ ∈ σ G ( H Υ ) by Lemma 2.4(i)(a). Weconclude from (2.56) and (2.92) that √ k ψ λ − ψ ν k ≤ (cid:13)(cid:13)(cid:13) ψ λ − ϕ ( a ) x (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ϕ ( a ) x − φ e ν (cid:13)(cid:13)(cid:13) + k φ e ν − ψ ν k (2.100) ≤ C d,ε L q ℓ − ( θ − d − ) + 2 C d,ε L d +2 q ℓ − ( θ − d ) < , a contradiction. igensystem Bootstrap Multiscale Analysis Lemma 2.7.
Given Λ L = Λ L ( x ) , x ∈ R d , let Υ be a ♯ -buffered subset of Λ L .Let G = G Υ and set E Λ L G ( ν ) = n λ ( a ) x ; a ∈ G , x ∈ Λ Λ L ,ℓ ♯ ℓ ( a ) , e λ ( a ) x = ν o ⊂ E Υ G ( ν ) for ν ∈ σ ( H Υ ) , (2.101) σ Λ L G ( H Υ ) = n ν ∈ σ ( H Υ ); E Λ L G ( λ ) = ∅ o ⊂ σ G ( H Υ ) . The following holds for sufficiently large ℓ : (i) Let ( ψ, λ ) be an eigenpair for H Λ L such that for all ν ∈ σ Λ L G ( H Υ ) ∪ σ B ( H Υ ) , | λ − ν | ≥ ( L − q if ♯ is θ -PL or m ∗ -ML e − L β if ♯ is s -SEL or m -LOC . (2.102) Then for all y ∈ Υ Λ L , ℓ ♯ : (a) If ♯ is θ -PL, we have | ψ ( y ) | ≤ C d,ε L d +2 q ℓ − ( θ − d ) | ψ ( v ) | for some v ∈ ∂ Λ L , ℓ ′ Υ . (2.103)(b) If ♯ is s -SEL, we have | ψ ( y ) | ≤ e − c ℓ s | ψ ( v ) | for some v ∈ ∂ Λ L , ℓ ′ Υ , (2.104) where c = c ( ℓ ) ≥ − C d,ε L β ℓ − s . (2.105)(c) If ♯ is m ∗ -ML, we have | ψ ( y ) | ≤ e − m ∗ ℓ τ | ψ ( v ) | for some v ∈ ∂ Λ L , ℓ τ Υ , (2.106) where m ∗ = m ∗ ( ℓ ) ≥ m ∗ − C d,ε γ q log ℓℓ τ . (2.107)(d) If ♯ is m -LOC, we have | ψ ( y ) | ≤ e − m ℓ τ | ψ ( v ) | for some v ∈ ∂ Λ L , ℓ τ Υ , (2.108) where m = m ( ℓ ) ≥ m − C d,ε ℓ γβ − τ . (2.109)(ii) Let Λ L be polynomially level spacing for H if ♯ is θ -PL or m ∗ -ML, level spac-ing for H if ♯ is s -SEL or m -LOC. Let { ( ψ λ , λ ) } λ ∈ σ ( H Λ L ) be an eigensystemfor H Λ L , and set (recalling (2.90) ) σ Υ ( H Λ L ) = { e ν ; ν ∈ σ B ( H Υ ) } ⊂ σ ( H Λ L ) \ σ G ( H Λ L ) . (2.110) Then the condition (2.102) is satisfied for all λ ∈ σ ( H Λ L ) \ ( σ G ( H Λ L ) ∪ σ Υ ( H Λ L )) , so for all y ∈ Υ Λ L , ℓ ♯ | ψ λ ( y ) | ≤ C d,ε L d +2 q ℓ − ( θ − d ) | ψ ( v ) | if ♯ is θ -PL e − c ℓ s | ψ ( v ) | if ♯ is s -SEL e − m ∗ ℓ τ | ψ ( v ) | if ♯ is m ∗ -ML e − m ℓ τ | ψ ( v ) | if ♯ is m -LOC (2.111) igensystem Bootstrap Multiscale Analysis for some v ∈ ∂ Λ L , ℓ ♯ Υ .Proof. Let { ( ϑ ν , ν ) } ν ∈ σ ( H Υ ) be an eigensystem for H Υ . For ν ∈ σ G ( H Υ ) we fix λ ( a ν ) x ν ∈ E Υ G ( ν ), where a ν ∈ G , x ν ∈ Λ Υ ,ℓ ′ ℓ ( a ν ). If ν ∈ σ Λ L G ( H Υ ), we choose λ ( a ν ) x ν ∈ E Λ L G ( ν ), thus x ν ∈ Λ Λ L ,ℓ ′ ℓ ( a ν ). If ν ∈ σ G ( H Υ ) \ σ Λ L G ( H Υ ) we have x ν ∈ Λ Υ ,ℓ ′ ℓ ( a ν ) \ Λ Λ L ,ℓ ′ ℓ ( a ν ).Given y ∈ Υ, we have (see (2.10)) ψ ( y ) = X ν ∈ σ (Υ) ϑ ν ( y ) h ϑ ν , ψ i (2.112)= X ν ∈ σ Λ L G ( H Υ ) ∪ σ B ( H Υ ) ϑ ν ( y ) h ϑ ν , ψ i + X ν ∈ σ G ( H Υ ) \ σ Λ L G ( H Υ ) ϑ ν ( y ) h ϑ ν , ψ i . Let ( ψ, λ ) be an eigenpair for H Λ L satisfying (2.102). 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 u , ψ i . (2.113)It follows from (2.102) and [EK1, Lemma 3.2] that | ϑ ν ( y ) h ϑ ν , ψ i| ≤ L q ε | ϑ ν ( y ) | X v ∈ ∂ Λ L ex Υ X v ′ ∈ ∂ Λ L in Υ , | v ′ − v | =1 | ϑ ν ( v ′ ) | | ψ ( v ) | (2.114) ≤ εL q + d 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 ), (2.88) givesmax u ∈ ∂ Λ L in Υ | ϑ ν ( u ) | ≤ C d,ε L q ℓ − ( θ − d ) . (2.115)If ν ∈ σ Λ L G ( H Υ ), it follows from (2.56) and (1.6), thatmax u ∈ ∂ Λ L in Υ | ϑ ν ( u ) | ≤ max u ∈ ∂ Λ L in (cid:16)(cid:12)(cid:12)(cid:12) ϑ ν ( u ) − ϕ ( a ν ) x ν (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ϕ ( a ν ) x ν (cid:12)(cid:12)(cid:12)(cid:17) (2.116) ≤ C d,ε L q ℓ − ( θ − d − ) + ℓ − θ ≤ C d,ε L q ℓ − ( θ − d − ) ≤ C d,ε L q ℓ − ( θ − d ) . Therefore (recalling (2.38)), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ν ∈ σ Λ L G ( H Υ ) ∪ σ B ( H Υ ) ϑ ν ( y ) h ϑ ν , ψ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ dεL d + q (cid:16) C d,ε L q ℓ − ( θ − d ) (cid:17) | ψ ( v ) | (2.117) ≤ C d,ε L d +2 q ℓ − ( θ − d ) | ψ ( v ) | , for some v ∈ ∂ Λ L ex Υ. igensystem Bootstrap Multiscale Analysis ν ∈ σ G ( H Υ ) \ σ Λ L G ( H Υ ), we have x ν ∈ Λ Υ ,ℓ ′ ℓ ( a ν ) \ Λ Λ L ,ℓ ′ ℓ ( a ν ), thusdist( x ν , Υ \ Λ ℓ ( a ν )) > ℓ ′ and dist( x ν , Λ L \ Λ ℓ ( a ν )) ≤ ℓ ′ , (2.118)and hence there is u ∈ Λ L \ Υ such that k x ν − u k ≤ ℓ ′ . We suppose y ∈ Υ Λ L , ℓ ′ ,then k y − u k > ℓ ′ . Therefore k x ν − y k ≥ k y − u k − k x ν − u k > ℓ ′ − ℓ ′ = ℓ ′ . (2.119)Thus it follows from (2.56) and (1.6) that | ϑ ν ( u ) | ≤ (cid:12)(cid:12)(cid:12) ϑ ν ( u ) − ϕ ( a ν ) x ν (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ϕ ( a ν ) x ν (cid:12)(cid:12)(cid:12) ≤ C d,ε L q ℓ − ( θ − d − ) + ℓ − θ (2.120) ≤ C d,ε L q ℓ − ( θ − d − ) . Therefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ν ∈ σ G ( H Υ ) \ σ Λ L G ( H Υ ) ϑ ν ( y ) h ϑ ν , ψ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C d,ε L q ( L + 1) d ℓ − ( θ − d − ) | ψ ( v ) | , (2.121)for some v ∈ Υ.Combining (2.112), (2.117) and (2.121), we conclude that for all y ∈ Υ Λ L , ℓ ′ , | ψ ( y ) | ≤ C d,ε L d +2 q ℓ − ( θ − d ) | ψ ( v ) | , (2.122)for some v ∈ Υ ∪ ∂ Λ L ex Υ. If v ∈ Υ Λ L , ℓ ′ we repeat the procedure to estimate | ψ ( v ) | . Since we can suppose ψ ( y ) = 0 without loss of generality, the proceduremust stop after finitely many times, and at that time we must have (2.103).Now let Λ L be polynomially level spacing. If λ σ G ( H Λ L ), it follows fromLemma 2.4(i)(c) that (2.65) holds for all a ∈ G . If λ σ Υ ( H Λ L ), using theargument in (2.75), with (2.91) instead of (2.55), we get | λ − ν | ≥ L − q for all ν ∈ σ B ( H Υ ). Therefore we have (2.102), which implies (2.103). The following lemma gives the probability estimates for polynomially level spacingand level spacing.
Lemma 3.1.
Let H ε,ω be the Anderson model. Let Θ ⊂ Z d and L > . Then, forall ε ≤ ε , P { Θ is L -polynomially level spacing for H } ≥ − Y ε L − (2 α − q | Θ | , (3.1) and P { Θ is L -level spacing for H } ≥ − Y ε e − (2 α − L β | Θ | , (3.2) where Y ε = 2 α − e K (diam supp µ + 2 dε + 1) , (3.3) with e K = K if α = 1 and e K = 8 K if α ∈ (cid:0) , (cid:1) . Lemma 3.1 follows from [EK1, Lemma 2.1] and its proof. (Also see [KlM,Lemma 2].) igensystem Bootstrap Multiscale Analysis In this section, we fix θ > (cid:16) α − + (cid:17) d and 0 < ξ <
1. (Note that Proposition 4.1is independent of ξ .) We will omit the dependence on θ and ξ from the notation.We denote the complementary event of an event E by E c . Proposition 4.1.
Fix ε > , Y ≥ , and P < (2 Y ) − d . There exists a finitescale L ( ε , Y ) with the following property: Suppose for some scale L ≥ L ( ε , Y ) ,and < ε ≤ ε we have inf x ∈ R d P { Λ L ( x ) is θ -polynomially localizing for H ε,ω } ≥ − P . (4.1) Then, setting L k +1 = Y L k for k = 0 , , . . . , there exists K = K ( Y, L , P ) ∈ N such that inf x ∈ R d P { Λ L k ( x ) is θ -polynomially localizing for H ε,ω } ≥ − L − pk for k ≥ K . (4.2)Proposition 4.1 follows from the following induction step for the multiscaleanalysis. Lemma 4.2.
Fix ε > , Y ≥ , and P ≤ . Suppose for some scale ℓ and < ε ≤ ε we have inf x ∈ R d P { Λ ℓ ( x ) is θ -polynomially localizing for H ε,ω } ≥ − P. (4.3) Then, if ℓ is sufficiently large, for L = Y ℓ we have inf x ∈ R d P { Λ L ( x ) is θ -polynomially localizing for H ε,ω } ≥ − (cid:0) (2 Y ) d P + L − p (cid:1) . (4.4) Proof.
We fix 0 < ε ≤ ε and suppose (4.3) for some scale ℓ . Let Λ L = Λ L ( x ),where x ∈ R d , and let C L,ℓ = C L,ℓ ( x ) be the suitable ℓ -cover of Λ L . For N ∈ N ,let B N denote the event that there exist at most N disjoint boxes in C L,ℓ that arenot θ -PL for H ε,ω . Using (4.3), (2.9) and the fact that events on disjoint boxesare independent, if N = 1 we have P {B cN } ≤ (cid:0) Lℓ (cid:1) ( N +1) d P N +1 = (2 Y ) ( N +1) d P N +1 = (2 Y ) d P . (4.5)We now fix ω ∈ B N . There exists A N = A N ( ω ) ∈ Ξ L,ℓ = Ξ
L,ℓ ( x ), with |A N | ≤ N and k a − b k ≥ ρℓ (i.e., Λ ℓ ( a ) ∩ Λ ℓ ( b ) = ∅ ) if a, b ∈ A N , a = b , such thatfor all a ∈ Ξ L,ℓ with dist( a, A N ) ≥ ρℓ (i.e., Λ ℓ ( a ) ∩ Λ ℓ ( b ) = ∅ for all b ∈ A N ),Λ ℓ ( a ) is a ♯ box for H ε,ω ( ♯ stands for θ -PL). In other words, a ∈ Ξ L,ℓ \ [ b ∈A N Λ R (2 ρ +1) ℓ ( a ) = ⇒ Λ ℓ ( a ) is a ♯ box for H ε,ω . (4.6) igensystem Bootstrap Multiscale Analysis { Λ ℓ ( b ) } b ∈A N into ♯ -buffered subsets of Λ L , we considergraphs G i = (Ξ L,ℓ , E i ), i = 1 ,
2, both having Ξ
L,ℓ as the set of vertices, with setsof edges given by E = {{ a, b } ∈ Ξ L,ℓ ; k a − b k = ρℓ } (4.7)= {{ a, b } ∈ Ξ L,ℓ ; a = b and Λ ℓ ( a ) ∩ Λ ℓ ( b ) = ∅} , E = {{ a, b } ∈ Ξ L,ℓ ; either k a − b k = 2 ρℓ or k a − b k = 3 ρℓ } = {{ a, b } ∈ Ξ L,ℓ ; Λ ℓ ( a ) ∩ Λ ℓ ( b ) = ∅ and Λ (2 ρ +1) ℓ ( a ) ∩ Λ (2 ρ +1) ℓ ( b ) = ∅} . 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 ). Note that R ∈ { , , . . . , N } , R X r =1 | Φ r | = |A N | ≤ N, and diam Φ r ≤ ρℓ ( | Φ r | − . (4.8)Set e Φ r = Ξ L,ℓ ∩ [ a ∈ Φ r Λ R (2 ρ +1) ℓ ( a ) = { a ∈ Ξ L,ℓ ; dist( a, Φ r ) ≤ ρℓ } , (4.9)and note that ne Φ r o Rr =1 is a collection of disjoint, G -connected subsets of Ξ L,ℓ ,such thatdiam e Φ r ≤ diam Φ r + 2 ρℓ ≤ ρℓ (3 | Φ r | −
1) and dist( e Φ r , e Φ e r ) ≥ ρℓ, r = e r. (4.10)Moreover, (4.6) gives a ∈ G = G ( ω ) = Ξ L,ℓ \ R [ r =1 e Φ r = ⇒ Λ ℓ ( a ) is a ♯ box for H ε,ω . (4.11)For Ψ ⊂ Ξ L,ℓ , we define the exterior boundary of Ψ in the graph G by ∂ G ex Ψ = { a ∈ Ξ L,ℓ ; dist( a, Ψ) = ρℓ } . (4.12)It follows from (4.11) that Λ ℓ ( a ) is ♯ for H ε,ω for all a ∈ ∂ G ex e Φ r , r = 1 , , . . . , R .Set Ψ = Ψ ∪ ∂ G ex Ψ, and set, for r = 1 , , . . . , R ,Υ (0) r = Υ (0) r ( ω ) = [ a ∈ e Φ r Λ ℓ ( a ) , (4.13)Υ r = Υ r ( ω ) = Υ (0) r ∪ [ a ∈ ∂ G e Φ r Λ ℓ ( a ) = [ a ∈ e Φ r Λ ℓ ( a ) . Each Υ r , r = 1 , , . . . , R , satisfies all the requirements to be a θ -PL-buffered subsetof Λ L with G Υ r = ∂ G ex e Φ r (see Definition 2.5), except that we do not know if Υ r is L -polynomially level spacing for H ε,ω . (Note that the sets { Υ (0) r } Rr =1 are disjoint,but the sets { Υ r } Rr =1 are not necessarily disjoint.) Note also thatdiam e Φ r ≤ diam e Φ r + 2 ρℓ ≤ ρℓ (3 | Φ r | + 1) , (4.14) igensystem Bootstrap Multiscale Analysis r ≤ diam e Φ r + ℓ ≤ ρℓ (3 | Φ r | + 1) + ℓ ≤ ℓ | Φ r | , (4.15)thus R X r =1 diam Υ r ≤ ℓN. (4.16)We can arrange for { Υ r } Rr =1 to be a collection of θ -PL-buffered subsets of Λ L as follows. It follows from Lemma 3.1 that for any Θ ⊂ Λ L we have P { Θ is L -polynomially level spacing for H ε,ω } ≥ − Y ε e − (2 α − L β ( L + 1) d . (4.17)Given a G -connected subset Φ of Ξ L,ℓ , let Υ(Φ) ⊂ Λ L be constructed from Φ asin (4.13). Set F N = N [ r =1 F ( r ) , where F ( r ) = { Φ ⊂ Ξ L,ℓ ; Φ is G -connected and | Φ | = r } . (4.18)Let F ( r, a ) = { Φ ∈ F r ; a ∈ Φ } for a ∈ Ξ L,ℓ , and note that each vertex in the graph G has less than d (3 d − + 4 d − ) ≤ d d nearest neighbors , we have |F ( r, a ) | ≤ ( r − d d ) r − = ⇒ |F ( r ) | ≤ ( L + 1) d ( r − d d ) r − (4.19)= ⇒ |F N | ≤ ( L + 1) d N !( d d ) N − . Let S N denote the event that the box Λ L and the subsets { Υ(Φ) } Φ ∈F N are all L -polynomially level spacing for H ε,ω , using (4.17) and (4.19), if N = 1 we have P {S cN } ≤ Y ε (cid:0) L + 1) d N !( d d ) N − (cid:1) ( L + 1) d ( L + 1) d L − (2 α − q < L − p (4.20)for sufficiently large L since p < (2 α − q − d .Let E N = B N ∩ S N . Combining (4.5) and (4.20), we conclude that if N = 1, P {E N } > − (cid:0) (2 Y ) d P + L − p (cid:1) . (4.21)To finish the proof we need to show that for all ω ∈ E N the box Λ L is θ -PL for H ε,ω .We fix ω ∈ E N . Then we have (4.11), Λ L is polynomially level spacing for H ε,ω ,and the subsets { Υ r } Rr =1 constructed in (4.13) are θ -PL-buffered subsets of Λ L for H ε,ω . It follows from (2.8) and Definition 2.5(iii) thatΛ L = ( [ a ∈G Λ Λ L , ℓ ℓ ( a ) ) ∪ ( R [ r =1 Υ Λ L , ℓ r ) . (4.22)We omit ε and ω from the notation since they are now fixed. Let { ( ψ λ , λ ) } λ ∈ σ ( H Λ L ) be an eigensystem for H Λ L . For a ∈ G , let n ( ϕ ( a ) x , λ ( a ) x ) o x ∈ Λ ℓ ( a ) be a θ -polynomially igensystem Bootstrap Multiscale Analysis 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 = ne ν ( r ) ; ν ( r ) ∈ σ B ( H Υ r ) o ⊂ σ ( H Λ L ) \ σ G ( H Λ L ) , (4.23)where e ν ( r ) is given in (2.90), which also gives σ Υ r ( H Λ L ) ⊂ σ ( H Λ L ) \ σ G Υ r ( H Λ L ),but the 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 ) . (4.24)We claim σ ( H Λ L ) = σ G ( H Λ L ) ∪ σ B ( H Λ L ) . (4.25)To do this, we assume λ ∈ σ G \ ( σ G ( H Λ L ) ∪ σ B ( H Λ L )). Since Λ L is polynomiallylevel spacing for H , Lemma 2.4(ii)(c) gives | ψ λ ( y ) | ≤ C d,ε L q ℓ − ( θ − d ) for all y ∈ [ a ∈G Λ Λ L , ℓ ′ ℓ ( a ) , (4.26)and Lemma 2.7(ii) gives | ψ λ ( y ) | ≤ C d,ε L d +2 q ℓ − ( θ − d ) for all y ∈ R [ r =1 Υ Λ L , ℓ ′ r . (4.27)Using (4.22) and θ − d > γ (cid:0) d + 2 q (cid:1) > d + 2 q , we conclude that1 = k ψ λ ( y ) k ≤ C d,ε L d +2 q ℓ − ( θ − d ) ( L + 1) d < ℓ , a contradiction. This establishes the claim.We now index the eigenvalues and eigenvectors of H Λ L by sites in Λ L usingHall’s Marriage Theorem, which states a necessary and sufficient condition for theexistence of a perfect matching in a bipartite graph. (See [EK1, Appendix C] and[BuDM, Chapter 2].) We consider the bipartite graph G = (Λ L , σ ( H Λ L ); E ), wherethe edge set E ⊂ Λ L × σ ( H Λ L ) is defined as follows. For each λ ∈ σ G ( H Λ L ) we fix λ ( a λ ) x λ ∈ E Λ L G ( λ ), and set (recall (2.86) and (2.19)) N ( x ) = ( { λ ∈ σ G ( H Λ L ); k x λ − x k < ℓ ♯ } for x ∈ Λ L \ S Rr =1 b Υ r ∅ for x ∈ S Rr =1 b Υ r . (4.29)We define N ( x ) = N ( x ) for x ∈ Λ L \ S Rr =1 c Υ ′ r σ Υ ( H Λ L ) for x ∈ b Υ r , r = 1 , , . . . , R N ( x ) ∪ σ Υ ( H Λ L ) for x ∈ c Υ ′ r , \ b Υ r , r = 1 , , . . . , R , (4.30)and let E = { ( x, λ ) ∈ Λ L × σ ( H Λ L ); λ ∈ N ( x ) } . N ( x ) was defined to ensure | ψ λ ( x ) | ≪ λ
6∈ N ( x ). This can be seen asfollows: igensystem Bootstrap Multiscale Analysis • If x ∈ Λ L and λ ∈ σ G ( H Λ L ) \ N ( x ), we have λ = e λ ( a λ ) x λ with k x λ − x k ≥ ℓ ′ ,so, using (1.6) and (2.56), | ψ λ ( x ) | ≤ (cid:12)(cid:12)(cid:12) ϕ ( a λ ) x λ ( x ) (cid:12)(cid:12)(cid:12) + (cid:13)(cid:13)(cid:13) ϕ ( a λ ) x λ − ψ λ (cid:13)(cid:13)(cid:13) ≤ ℓ − Θ + 2 C d,ε L q ℓ − ( θ − d − ) (4.31) ≤ C d,ε L q ℓ − ( θ − d − ) . • If x ∈ Λ L \ c Υ ′ r and λ ∈ σ Υ r ( H Λ L ), then λ = e ν ( r ) for some ν ( r ) ∈ σ B ( H Υ ),and, using (2.88) and (2.92), (Note φ ν ( r ) ( x ) = 0 if x Υ r .) | ψ λ ( x ) | ≤ | φ ν ( r ) ( x ) | + k φ ν ( r ) ( x ) − ψ λ k ≤ C d,ε L q ℓ − ( θ − d ) + 2 C d,ε L d +2 q ℓ − ( θ − d ) (4.32) ≤ C d,ε L d +2 q ℓ − ( θ − d ) . Therefore for all x ∈ Λ L and λ ∈ σ ( H Λ L ) \ N ( x ) we have | ψ λ ( x ) | ≤ C d,ε L d +2 q ℓ − ( θ − d ) . (4.33)Since | Λ L | = | σ ( H Λ L ) | , to apply Hall’s Marriage Theorem we only need toverify | Θ | ≤ |N (Θ) | , where N (Θ) = S x ∈ Θ N ( x ) for Θ ⊂ Λ L . For Θ ⊂ Λ L , let Q Θ be the orthogonal projection onto the span of { ψ λ ; λ ∈ N (Θ) } . If λ
6∈ N (Θ), forall x ∈ Θ we have (4.33), thus k (1 − Q Θ ) χ Θ k ≤ | Λ L | | Θ | C d,ε L d +2 q ℓ − ( θ − d ) (4.34) ≤ ( L + 1) d C d,ε L d +2 q ℓ − ( θ − d ) < , for sufficiently large ℓ since θ − d > γ (cid:0) d + 2 q (cid:1) > d + 2 q , so it follows from[EK1, Lemma A.1] that | Θ | = tr χ Θ ≤ tr Q Θ = |N (Θ) | . (4.35)Using Hall’s Marriage Theorem, we conclude that there exists a bijection x ∈ Λ L λ x ∈ σ ( H Λ L ) , where λ x ∈ N ( x ) . (4.36)We set ψ x = ψ λ x for all x ∈ Λ L .To finish the proof we need to show that { ( ψ x , λ x ) } x ∈ Λ L is a θ -polynomiallylocalized eigensystem for Λ L . We fix N = 1, x ∈ Λ L , take y ∈ Λ L , and considerseveral cases:(i) Suppose λ x ∈ σ G (Λ L ). Then x ∈ Λ ℓ ( a λ x ) with a λ x ∈ G , and λ x ∈ σ { a λx } ( H Λ L ).In view of (4.22) we consider two cases:(a) If y ∈ Λ Λ L , ℓ ℓ ( a ) for some a ∈ G and k y − x k ≥ ℓ , we must haveΛ ℓ ( a λ x ) ∩ Λ ℓ ( a ) = ∅ , so it follows from (2.70) that λ x σ { a } ( H Λ L ), and(2.66) gives | ψ x | ≤ C d,ε L q ℓ − ( θ − d ) | ψ x ( y ) | for some y ∈ ∂ Θ , ℓ ′ Λ ℓ ( a ) . (4.37) igensystem Bootstrap Multiscale Analysis y ∈ Υ Λ L , ℓ , and k y − x k ≥ ℓ + diam Υ , we must have Λ ℓ ( a λ x ) ∩ Υ = ∅ , so it follows from (2.70) that λ x σ G Υ1 ( H Λ L ), and clearly λ x σ Υ ( H Λ L ) in view of (4.23). Thus Lemma 2.7(ii) gives | ψ x ( y ) | ≤ C d,ε L d +2 q ℓ − ( θ − d ) | ψ x ( v ) | for some v ∈ ∂ Λ L , ℓ ′ Υ . (4.38)(ii) Suppose λ x σ G (Λ L ). Then it follows from (4.25) that we must have λ x ∈ σ Υ ( H Λ L ). If y ∈ Λ Λ L , ℓ ℓ ( a ) for some a ∈ G , and k y − x k ≥ ℓ + diam Υ , wemust have Λ ℓ ( a ) ∩ Υ = ∅ , and (2.66) gives (4.37).Now we fix x ∈ Λ L , and take y ∈ Λ L such that k y − x k ≥ L ′ . Suppose | ψ x ( y ) | > | ψ x ( y ) | using either (4.37) or(4.38) repeatedly, as appropriate, stopping when we get too close to x so we are notin any case described above. (Note that this must happen since | ψ x ( y ) | > C d,ε L d +2 q ℓ − ( θ − d ) < L = Y ℓ , we get | ψ x ( y ) | ≤ (cid:16) C d,ε L q ℓ − ( θ − d ) (cid:17) n ( Y ) , (4.39)where n ( Y ) is the number of times we used (4.37). We have n ( Y )( ℓ + 1) + diam Υ + 2 ℓ ≥ L ′ . (4.40)Thus, using (4.16), we have n ( Y ) ≥ ℓ +1 ( L ′ − ℓ − ℓ ) ≥ ℓℓ +1 (cid:0) Y − (cid:1) ≥ . (4.41)for sufficiently large ℓ since Y ≥ | ψ x ( y ) | ≤ (cid:16) C d,ε Y q ℓ − ( θ − d − q ) (cid:17) ≤ L − θ , (4.42)for sufficiently large ℓ since 2( θ − d − q ) = θ + ( θ − d − q ) > θ .We conclude that { ( ψ x , λ x ) } x ∈ Λ L is a θ -polynomially localized eigensystem forΛ L , so the box Λ L is θ -polynomially localizing for H ε,ω . Proof of Proposition 4.1.
We assume (4.1) and set L k +1 = Y L k for k = 0 , , . . . .We set P k = sup x ∈ R d P { Λ L k ( x ) is not θ -polynomially localizing for H ε,ω } for k = 1 , , . . . . (4.43)Then by Lemma 4.2, we have P k +1 ≤ (2 Y ) d P k + L − pk +1 for k = 0 , , . . . (4.44)If P k ≤ L − pk for some k ≥
0, we have P k +1 ≤ (2 Y ) d L − pk + L − pk +1 ≤ (2 Y ) d +2 p L − pk +1 + L − pk +1 ≤ L − pk +1 (4.45) igensystem Bootstrap Multiscale Analysis L sufficiently large. Therefore to finish the proof, we need to show that K = inf { k ∈ N ; P k ≤ L − pk } < ∞ . (4.46)It follows from (4.44) that for any 1 ≤ k < K , P k ≤ (2 Y ) d P k − + L − pk < (2 Y ) d P k − + P k , (4.47)so 2(2 Y ) d P k < (cid:0) Y ) d P k − (cid:1) . (4.48)Therefore for 1 ≤ k < K , we have2 d +1 Y − ( kp − d ) L − p = 2(2 Y ) d L − pk < Y ) d P k < (cid:0) Y ) d P (cid:1) k . (4.49)Since 2(2 Y ) d P <
1, (4.49) cannot be satisfied for large k . We conclude that K < ∞ . Proposition 4.3.
Fix ε > . Suppose for some scale ℓ and < ε ≤ ε we have inf x ∈ R d P { Λ ℓ ( x ) is θ -polynomially localizing for H ε,ω } ≥ − ℓ − p . (4.50) Then, if ℓ is sufficiently large, for L = ℓ γ we have inf x ∈ R d P { Λ L ( x ) is m ∗ -mix localizing for H ε,ω } ≥ − L − p , (4.51) where m ∗ ≥ (cid:0) d + q (cid:1) L − (1 − τ + γ ) log L. (4.52) Proof.
We follow the proof of Lemma 4.2. For N ∈ N , let B N , S N and E N as inthe proof of Lemma 4.2. Using (4.50), (2.9) and the fact that events on disjointboxes are independent, if N = 1 we have, P {B cN } ≤ (cid:0) Lℓ (cid:1) d ℓ − p = 2 d ℓ − p − d ( γ − < ℓ − γ p = L − p (4.53)for all ℓ sufficiently large since 1 < γ < pp +2 d . Also, using (4.17) and (4.19),if N = 1 we have, P {S cN } ≤ (cid:0) L + 1) d (cid:1) Y ε ( L + 1) d L − (2 α − q < L − p (4.54)for sufficiently large L , since p < (2 α − q − d . Combining (4.53) and (4.54), weconclude that P {E N } > − L − p . (4.55)To finish the proof we need to show that for all ω ∈ E N the box Λ L is m ∗ -mixlocalizing for H ε,ω , where m ∗ is given in (4.52). Following the proof of Lemma 4.2,we get (4.25) and obtain an eigensystem { ( ψ x , λ x ) } x ∈ Λ L for H Λ L using Hall’sMarriage Theorem. To finish the proof we need to show that { ( ψ x , λ x ) } x ∈ Λ L is an igensystem Bootstrap Multiscale Analysis m ∗ -localized eigensystem for Λ L . We proceed as in the proof of Lemma 4.2. Wefix N = 1, x ∈ Λ L , and take y ∈ Λ L such that k y − x k ≥ L τ , we have n ( ℓ )( ℓ + 1) + diam Υ + 2 ℓ ≥ L τ . (4.56)where n ( ℓ ) is the number of times we used (4.37). Thus, using (4.16), we have n ( ℓ ) ≥ ℓ +1 ( L τ − ℓ − ℓ ) ≥ ℓℓ +1 (cid:0) ℓ γ τ − − (cid:1) ≥ ℓ γ τ − . (4.57)for sufficiently large ℓ . It follows from (4.39), | ψ x ( y ) | ≤ (cid:16) C d,ε ℓ − ( θ − d − γ q ) (cid:17) ℓ γ τ − (4.58) ≤ e − ( d + q ) L − (1 − τ + 1 γ (log L ) k y − x k , for sufficiently large ℓ .We conclude that { ( ψ x , λ x ) } x ∈ Λ L is an m ∗ -localized eigensystem for Λ L , where m ∗ is given in (4.52), so the box Λ L is m ∗ -mix localizing for H ε,ω . Proposition 4.4.
Fix ε > . There exists a finite scale L ( ε ) with the followingproperty: Suppose for some scale L ≥ L ( ε ) , < ε ≤ ε , and m ∗ ≥ L − κ where < κ < τ , we have inf x ∈ R d P { Λ L ( x ) is m ∗ -mix localizing for H ε,ω } ≥ − L − p . (4.59) Then, setting L k +1 = L γ k for k = 0 , , . . . , we have inf x ∈ R d P { Λ L k ( x ) is m ∗ -mix localizing for H ε,ω } ≥ − L − pk for k = 0 , , . . . . (4.60)Proposition 4.4 follows from the following induction step for the multiscaleanalysis. Lemma 4.5.
Fix ε > . Suppose for some scale ℓ , < ε ≤ ε , and m ∗ ≥ ℓ − κ ,where < κ < τ , we have inf x ∈ R d P { Λ ℓ ( x ) is m ∗ -mix localizing for H ε,ω } ≥ − ℓ − p . (4.61) Then, if ℓ is sufficiently large, for L = ℓ γ we have inf x ∈ R d P { Λ L ( x ) is M ∗ -mix localizing for H ε,ω } ≥ − L − p , (4.62) where M ∗ ≥ m ∗ (cid:16) − C d,ε γ qℓ − min { − τ ,γ τ − ,τ − κ } (cid:17) ≥ L − κ . (4.63) igensystem Bootstrap Multiscale Analysis Proof.
We follow the proof of Lemma 4.2. For N ∈ N , let B N denote the event thatthere do not exist two disjoint boxes in C L,ℓ that are not m ∗ -mix localizing for H ε,ω .Using (4.61), (2.9) and the fact that events on disjoint boxes are independent, if N = 1 we have P {B cN } ≤ (cid:0) Lℓ (cid:1) ( N +1) d ℓ − ( N +1) p = 2 d ℓ − (2 p − d ( γ − < ℓ − γ p = L − p (4.64)for all ℓ sufficiently large since 1 < γ < pp +2 d .We now fix ω ∈ B N , and proceed as in the proof of Lemma 4.2 with ♯ being m ∗ -ML. Then we have Υ r , r = 1 , , . . . , R such that each Υ r satisfies all therequirements to be an m ∗ -ML-buffered subset of Λ L with G Υ r = ∂ G ex e Φ r , except wedo not know if Υ r is L -polynomially level spacing for H ε,ω .Given a G -connected subset Φ of Ξ L,ℓ , let Υ(Φ) ⊂ Λ L be constructed from Φas in (4.13) with ♯ being m ∗ -ML. Let S N denote the event that the box Λ L and thesubsets { Υ(Φ) } Φ ∈F N are all L -polynomially level spacing for H ε,ω . Using (4.17)and (4.19), if N = 1 we have P {S c } ≤ (cid:16) (cid:0) Lℓ (cid:1) d (cid:17) Y ε ( L + 1) d L − (2 α − q < L − p (4.65)for sufficiently large L , since p < (2 α − q − d .Let E N = B N ∩ S N . Combining (4.64) and (4.65), we conclude that if N = 1, P {E N } > − L − p . (4.66)To finish the proof we need to show that for all ω ∈ E N the box Λ L is M ∗ -mixlocalizing for H ε,ω , where M ∗ is given in (4.63).We fix ω ∈ E N . Then we have (4.11), Λ L is polynomially level spacing for H ε,ω , and the subsets { Υ r } Rr =1 constructed in (4.13) are m ∗ -ML-buffered subsetof Λ L for H ε,ω . We proceed as in the proof of Lemma 4.2. To claim (4.25), weassume λ ∈ σ G \ ( σ G ( H Λ L ) ∪ σ B ( H Λ L )). Since Λ L is polynomially level spacing for H , Lemma 2.4(ii)(c) gives | ψ λ ( y ) | ≤ e − m ∗ ℓ τ for all y ∈ [ a ∈G Λ Λ L , ℓ τ ℓ ( a ) , (4.67)and Lemma 2.7(ii) gives | ψ λ ( y ) | ≤ e − m ∗ ℓ τ for all y ∈ R [ r =1 Υ Λ L , ℓ τ r . (4.68)Using (4.22), we conclude that (note m ∗ ≤ m ∗ )1 = k ψ λ ( y ) k ≤ e − m ∗ ℓ τ ( L + 1) d < , (4.69)a contradiction. This establishes the claim.To index the eigenvalues and eigenvectors of H Λ L by sites in Λ L , we define N ( x ) as in (4.30) and proceed as in the proof of Lemma 4.2. We have: igensystem Bootstrap Multiscale Analysis • If x ∈ Λ L and λ ∈ σ G ( H Λ L ) \ N ( x ), we have λ = e λ ( a λ ) x λ with k x λ − x k ≥ ℓ τ ,so, using (1.8) and (2.60), | ψ λ ( x ) | ≤ (cid:12)(cid:12)(cid:12) ϕ ( a λ ) x λ ( x ) (cid:12)(cid:12)(cid:12) + (cid:13)(cid:13)(cid:13) ϕ ( a λ ) x λ − ψ λ (cid:13)(cid:13)(cid:13) ≤ e − m ∗ ℓ τ + 2e − m ∗ ℓ τ L q ≤ − m ℓ τ L q . (4.70) • If x ∈ Λ L \ c Υ ′ r and λ ∈ σ Υ r ( H Λ L ), then λ = e ν ( r ) for some ν ( r ) ∈ σ B ( H Υ r ),and, using (2.88) and (2.96), (Note φ ν ( r ) ( x ) = 0 if x Υ r .) | ψ λ ( x ) | ≤ | φ ν ( r ) ( x ) | + k φ ν ( r ) ( x ) − ψ λ k ≤ e − m ∗ ℓ τ + 2e − m ∗ ℓ τ L q ≤ − m ∗ ℓ τ L q . (4.71)Therefore for all x ∈ Λ L and λ ∈ σ ( H Λ L ) \ N ( x ) we have | ψ λ ( x ) | ≤ − m ∗ ℓ τ L q ≤ e − m ∗ ℓ τ . (4.72)If λ
6∈ N (Θ), for all x ∈ Θ we have (4.72), thus k (1 − Q Θ ) χ Θ k ≤ | Λ L | | Θ | e − m ∗ ℓ τ ≤ ( L + 1) d e − m ∗ ℓ τ < . (4.73)Following the proof of Lemma 4.2, we can apply Hall’s Marriage Theorem to obtainan eigensystem { ( ψ x , λ x ) } x ∈ Λ L for H Λ L .To finish the proof we need to show that { ( ψ x , λ x ) } x ∈ Λ L is an M ∗ -localizedeigensystem for Λ L , where M ∗ is given in (4.63). We fix N = 1, x ∈ Λ L , take y ∈ Λ L , and consider several cases:(i) Suppose λ x ∈ σ G (Λ L ). Then x ∈ Λ ℓ ( a λ x ) with a λ x ∈ G , and λ x ∈ σ { a λx } ( H Λ L ).In view of (4.22) we consider two cases:(a) If y ∈ Λ Λ L , ℓ ℓ ( a ) for some a ∈ G and k y − x k ≥ ℓ , we must haveΛ ℓ ( a λ x ) ∩ Λ ℓ ( a ) = ∅ , so it follows from (2.70) that λ x σ { a } ( H Λ L ), and(2.67) gives | ψ x | ≤ e − m ∗ k y − y k | ψ x ( y ) | for some y ∈ ∂ Θ ,ℓ e τ Λ ℓ ( a ) . (4.74)(b) If y ∈ Υ Λ L , ℓ , and k y − x k ≥ ℓ + diam Υ , we must have Λ ℓ ( a λ x ) ∩ Υ = ∅ , so it follows from (2.70) that λ x σ G Υ1 ( H Λ L ), and clearly λ x σ Υ ( H Λ L ) in view of (4.23). Thus Lemma 2.7(ii) gives | ψ x ( y ) | ≤ e − m ∗ ℓ τ | ψ x ( v ) | for some v ∈ ∂ Λ L , ℓ τ Υ . (4.75)(i) Suppose λ x σ G (Λ L ). Then it follows from (4.25) that we must have λ x ∈ σ Υ ( H Λ L ). If y ∈ Λ Λ L , ℓ ℓ ( a ) for some a ∈ G , and k y − x k ≥ ℓ + diam Υ , wemust have Λ ℓ ( a ) ∩ Υ = ∅ , and (2.67) gives (4.74).Now we fix x ∈ Λ L , and take y ∈ Λ L such that k y − x k ≥ L τ . Suppose | ψ x ( y ) | > | ψ x ( y ) | using either (4.74) or(4.75) repeatedly, as appropriate, stopping when we get too close to x so we are igensystem Bootstrap Multiscale Analysis | ψ x ( y ) | > − m ∗ ℓ τ < | ψ x ( y ) | ≤ e − m ∗ ( k y − x k− diam Υ − ℓ ) ≤ e − m ∗ ( k y − x k− ℓ ) (4.76) ≤ e − m ∗ k y − x k ( − ℓ − γ τ ) ≤ e M k y − x k , where we used (4.16) and took M ∗ = m ∗ (cid:0) − ℓ − γ τ (cid:1) ≥ (cid:16) m ∗ (cid:16) − ℓ τ − (cid:17) − C d,ε γ q log ℓℓ e τ (cid:17) (cid:0) − ℓ − γ τ (cid:1) (4.77) ≥ m ∗ (cid:16) − ℓ τ − − C d,ε γ qℓ κ − τ (cid:17) (cid:0) − ℓ − γ τ (cid:1) ≥ m ∗ (cid:16) − C d,ε γ qℓ − min { − τ ,γ τ − ,τ − κ } (cid:17) ≥ ℓ − κ ≥ ℓ − γ κ = L − κ for ℓ sufficiently large, where we used (2.29) and m ∗ ≥ ℓ − κ .We conclude that { ( ψ x , λ x ) } x ∈ Λ L is an M ∗ -localized eigensystem for Λ L , where M ∗ is given in (4.63), so the box Λ L is M ∗ -mix localizing for H ε,ω . Proof of Proposition 4.4.
We assume (4.59) and set L k +1 = L γ k for k = 0 , , . . . .If L is sufficiently large it follows from Lemma 4.5 by an induction argument thatinf x ∈ R d P { Λ L k ( x ) is m ∗ k -localizing for H ε,ω } ≥ − L − pk for k = 0 , , . . . , (4.78)where for k = 1 , , . . . we have m ∗ k ≥ m ∗ k − (cid:0) − C d,ε γ qL − ̺k − (cid:1) , with ̺ = min (cid:8) − τ , γ τ − , τ − κ (cid:9) . (4.79)Thus for all k = 1 , , . . . , taking L sufficiently large we get m ∗ k ≥ m ∗ k − Y j =0 (cid:16) − C d,ε γ qL − ̺γ j (cid:17) ≥ m ∗ ∞ Y j =0 (cid:16) − C d,ε γ qL − ̺γ j (cid:17) ≥ m ∗ , (4.80)finishing the proof of Proposition 4.4. Proposition 4.6.
Fix ε > , Y ≥ − s , and e P < (cid:0) Y ) ( ⌊ Y s ⌋ +1) d (cid:1) − ⌊ Y s ⌋ .There exists a finite scale L ( ε , Y ) with the following property: Suppose for somescale L ≥ L ( ε , Y ) and < ε ≤ ε we have inf x ∈ R d P { Λ L ( x ) is s -SEL for H ε,ω } ≥ − e P . (4.81) igensystem Bootstrap Multiscale Analysis Then, setting L k +1 = Y L k for k = 0 , , . . . , there exists K = K ( Y, L , e P ) ∈ N such that inf x ∈ R d P { Λ L k ( x ) is s -SEL for H ε,ω } ≥ − e − L ζk for k ≥ K . (4.82)Proposition 4.6 follows from the following induction step for the multiscaleanalysis. Lemma 4.7.
Fix ε > , Y ≥ − s , and ≤ P ≤ . Suppose for some scale ℓ and < ε ≤ ε we have inf x ∈ R d P { Λ ℓ ( x ) is s -SEL for H ε,ω } ≥ − P. (4.83) Then, if ℓ is sufficiently large, for L = Y ℓ we have inf x ∈ R d P { Λ L ( x ) is s -SEL for H ε,ω } ≥ − (cid:16) (2 Y ) ( ⌊ Y s ⌋ +1) d P ⌊ Y s ⌋ +1 + e − L ζ (cid:17) . (4.84) Proof.
We follow the proof of Lemma 4.2. For N ∈ N , let B N denote the eventthat there exist at most N disjoint boxes in C L,ℓ that are not s -SEL for H ε,ω .Using (4.83), (2.9) and the fact that events on disjoint boxes are independent, if N = ⌊ Y s ⌋ we have P {B c } ≤ (cid:0) Lℓ (cid:1) ( N +1) d P N +1 = (2 Y ) ( ⌊ Y s ⌋ +1) d P ⌊ Y s ⌋ +1 . (4.85)We now fix ω ∈ B N , and proceed as in the proof of Lemma 4.2 with ♯ being s -SEL. Then we have Υ r , r = 1 , , . . . , R such that each Υ r satisfies all the re-quirements to be an s -SEL-buffered subset of Λ L with G Υ r = ∂ G ex e Φ r , except wedo not know if Υ r is L -level spacing for H ε,ω .It follows from Lemma 3.1 that for any Θ ⊂ Λ L we have P { Θ is L -level spacing for H ε,ω } ≥ − Y ε e − (2 α − L β ( L + 1) d . (4.86)Given a G -connected subset Φ of Ξ L,ℓ , let Υ(Φ) ⊂ Λ L be constructed from Φ asin (4.13) with ♯ being s -SEL. Let S N denote the event that the box Λ L and thesubsets the subsets { Υ(Φ) } Φ ∈F N are all L -level spacing for H ε,ω . Using (4.86) and(4.19), if N = ⌊ Y s ⌋ we have P {S cN } ≤ Y ε (cid:0) L + 1) d N !( d d ) N − (cid:1) ( L + 1) d e − (2 α − L β < e − L ζ (4.87)for sufficiently large L , since ζ < β .Let E N = B N ∩ S N . Combining (4.85) and (4.87), we conclude that P {E N } > − (cid:16) (2 Y ) ( ⌊ Y s ⌋ +1) d P ⌊ Y s ⌋ +1 + e − L ζ (cid:17) . (4.88)To finish the proof we need to show that for all ω ∈ E N the box Λ L is s -SEL for H ε,ω . igensystem Bootstrap Multiscale Analysis ω ∈ E N . Then we have (4.11), Λ L is level spacing for H ε,ω , and thesubsets { Υ r } Rr =1 constructed in (4.13) are s -SEL-buffered subsets of Λ L for H ε,ω .We proceed as in the proof of Lemma 4.2. To claim (4.25), we assume λ ∈ σ G \ ( σ G ( H Λ L ) ∪ σ B ( H Λ L )). Since Λ L is level spacing for H , Lemma 2.4(ii)(c) gives | ψ λ ( y ) | ≤ e − c ℓ s for all y ∈ [ a ∈G Λ Λ L , ℓ ′ ℓ ( a ) , (4.89)and Lemma 2.7(ii) gives | ψ λ ( y ) | ≤ e − c ℓ s for all y ∈ R [ r =1 Υ Λ L , ℓ ′ r . (4.90)Using (4.22), we conclude that (note c ≤ c )1 = k ψ λ ( y ) k ≤ e − c ℓ s ( L + 1) d < , (4.91)a contradiction. This establishes the claim.To index the eigenvalues and eigenvectors of H Λ L by sites in Λ L , we define N ( x ) as in (4.30) proceed as in the proof of Lemma 4.2. We have: • If x ∈ Λ L and λ ∈ σ G ( H Λ L ) \ N ( x ), we have λ = e λ ( a λ ) x λ with k x λ − x k ≥ ℓ ′ ,so, using (1.7) and (2.58), | ψ λ ( x ) | ≤ (cid:12)(cid:12)(cid:12) ϕ ( a λ ) x λ ( x ) (cid:12)(cid:12)(cid:12) + (cid:13)(cid:13)(cid:13) ϕ ( a λ ) x λ − ψ λ (cid:13)(cid:13)(cid:13) ≤ e − ℓ s + 2e − c ℓ s e L β ≤ − c ℓ s e L β . (4.92) • If x ∈ Λ L \ c Υ ′ r and λ ∈ σ Υ r ( H Λ L ), then λ = e ν ( r ) for some ν ( r ) ∈ σ B ( H Υ r ),and, using (2.88) and (2.94), (Note φ ν ( r ) ( x ) = 0 if x Υ r .) | ψ λ ( x ) | ≤ | φ ν ( x ) | + k φ ν ( x ) − ψ λ k ≤ e − c ℓ s + 2e − c ℓ s e L β ≤ − c ℓ s e L β . (4.93)Therefore for all x ∈ Λ L and λ ∈ σ ( H Λ L ) \ N ( x ) we have | ψ λ ( x ) | ≤ − c ℓ s e L β ≤ e − c ℓ s . (4.94)If λ
6∈ N (Θ), for all x ∈ Θ we have (4.94), thus k (1 − Q Θ ) χ Θ k ≤ | Λ L | | Θ | e − c ℓ s ≤ ( L + 1) d e − c ℓ s < . (4.95)Following the proof of Lemma 4.2, we can apply Hall’s Marriage Theorem to obtainan eigensystem { ( ψ x , λ x ) } x ∈ Λ L for H Λ L .To finish the proof we need to show that { ( ψ x , λ x ) } x ∈ Λ L is an s -subexponentiallylocalized eigensystem for Λ L . We fix N = ⌊ Y s ⌋ , x ∈ Λ L , take y ∈ Λ L , and considerseveral cases:(i) Suppose λ x ∈ σ G (Λ L ). Then x ∈ Λ ℓ ( a λ x ) with a λ x ∈ G , and λ x ∈ σ { a λx } ( H Λ L ).In view of (4.22) we consider two cases: igensystem Bootstrap Multiscale Analysis y ∈ Λ Λ L , ℓ ℓ ( a ) for some a ∈ G and k y − x k ≥ ℓ , we must haveΛ ℓ ( a λ x ) ∩ Λ ℓ ( a ) = ∅ , so it follows from (2.70) that λ x σ { a } ( H Λ L ), and(2.66) gives | ψ x | ≤ e − c ℓ s | ψ x ( y ) | for some y ∈ ∂ Θ , ℓ ′ Λ ℓ ( a ) . (4.96)(b) If y ∈ Υ Λ L , ℓ r for some r ∈ { , , . . . , R } , and k y − x k ≥ ℓ + diam Υ r ,we must have Λ ℓ ( a λ x ) ∩ Υ r = ∅ , so it follows from (2.70) that λ x σ G Υ r ( H Λ L ), and clearly λ x σ Υ r ( H Λ L ) in view of (4.23). Thus Lemma 2.7(ii)gives | ψ x ( y ) | ≤ e − c ℓ s | ψ x ( v ) | for some v ∈ ∂ Λ L , ℓ ′ Υ r . (4.97)(ii) Suppose λ x σ G (Λ L ). Then it follows from (4.25) that we must have λ x ∈ σ Υ e r ( H Λ L ) for some e r ∈ { , , . . . , R } . In view of (4.22) we consider twocases:(a) If y ∈ Λ Λ L , ℓ ℓ ( a ) for some a ∈ G , and k y − x k ≥ ℓ + diam Υ e r , we musthave Λ ℓ ( a ) ∩ Υ e r = ∅ , and (2.66) gives (4.96).(b) If y ∈ Υ Λ L , ℓ r for some r ∈ { , , . . . , R } , and k y − x k ≥ diam Υ e r +diam Υ r , we must have r = e r . Thus Lemma 2.7(ii) gives (4.97).Now we fix x ∈ Λ L , and take y ∈ Λ L such that k y − x k ≥ L ′ . Suppose | ψ x ( y ) | > | ψ x ( y ) | using either (4.96) or(4.97) repeatedly, as appropriate, stopping when we get too close to x so we arenot in any case described above. (Note that this must happen since | ψ x ( y ) | > − c ℓ s < L = Y ℓ , then we get | ψ x ( y ) | ≤ (cid:16) e − c ℓ s (cid:17) n ( Y ) , (4.98)where n ( Y ) is the number of times we used (4.96). We have n ( Y )( ℓ + 1) + R X r =1 diam Υ r + 2 ℓ ≥ L ′ . (4.99)Thus, using (4.16), we have n ( Y ) ≥ ℓ +1 ( L ′ − ℓ ⌊ Y s ⌋ − ℓ ) ≥ ℓℓ +1 (cid:0) Y − Y s − (cid:1) ≥ Y s . (4.100)for sufficiently large ℓ since Y ≥ − s . It follows from (4.98), | ψ x ( y ) | ≤ (cid:16) e − c ℓ s (cid:17) Y s ≤ e − L s , (4.101)for sufficiently large ℓ .We conclude that { ( ψ x , λ x ) } x ∈ Λ L is an s -subexponentially localized eigensys-tem for Λ L , so the box Λ L is s -SEL for H ε,ω . igensystem Bootstrap Multiscale Analysis Proof of Proposition 4.6.
We assume (4.81) and set L k +1 = Y L k for k = 0 , , . . . .We set e P k = sup x ∈ R d P { Λ L k ( x ) is not s -SEL for H ε,ω } for k = 1 , , . . . . (4.102)Then by Lemma 4.7, we have e P k +1 ≤ (2 Y ) ( ⌊ Y s ⌋ +1) d e P ⌊ Y s ⌋ +1 k + e − L ζk +1 for k = 0 , , . . . (4.103)If e P k ≤ e − L ζk for some k ≥
0, we have e P k +1 ≤ (2 Y ) ( ⌊ Y s ⌋ +1) d (cid:16) e − L ζk (cid:17) ⌊ Y s ⌋ +1 + e − L ζk +1 (4.104) ≤ (2 Y ) ( ⌊ Y s ⌋ +1) d e − ⌊ Y s ⌋ +1 Y ζ L ζk +1 + e − L ζk +1 ≤ e − L ζk +1 for L sufficiently large, since ζ < s . Therefore to finish the proof, we need toshow that K = inf { k ∈ N ; e P k ≤ e − L ζk } < ∞ . (4.105)It follows from (4.103) that for any 1 ≤ k < K , e P k ≤ (2 Y ) ( ⌊ Y s ⌋ +1) d e P ⌊ Y s ⌋ +1 k − + e − L k + ζ < (2 Y ) ( ⌊ Y s ⌋ +1) d e P ⌊ Y s ⌋ +1 k − + e P k , (4.106)so (cid:16) Y ) ( ⌊ Y s ⌋ +1) d (cid:17) ⌊ Y s ⌋ e P k < (cid:18)(cid:16) Y ) ( N +1) d (cid:17) ⌊ Y s ⌋ e P k − (cid:19) ⌊ Y s ⌋ +1 . (4.107)For 1 ≤ k < K , since (cid:0) Y ) ( ⌊ Y s ⌋ +1) d (cid:1) ⌊ Y s ⌋ e P <
1, we have (cid:16) Y ) ( ⌊ Y s ⌋ +1) d (cid:17) ⌊ Y s ⌋ e − Y kζ L ζ = (cid:16) Y ) ( ⌊ Y s ⌋ +1) d (cid:17) ⌊ Y s ⌋ e − L ζk (4.108) < (cid:16) Y ) ( ⌊ Y s ⌋ +1) d (cid:17) ⌊ Y s ⌋ e P k < (cid:18)(cid:16) Y ) ( ⌊ Y s ⌋ +1) d (cid:17) ⌊ Y s ⌋ e P (cid:19) ( ⌊ Y s ⌋ +1) k ≤ (cid:18)(cid:16) Y ) ( ⌊ Y s ⌋ +1) d (cid:17) ⌊ Y s ⌋ e P (cid:19) Y ks . Since ζ < s , (cid:0) Y ) ( ⌊ Y s ⌋ +1) d (cid:1) ⌊ Y s ⌋ e P <
1, (4.108) cannot be satisfied for large k .We conclude that K < ∞ . Proposition 4.8.
Fix ε > . Suppose for some scale ℓ and < ε ≤ ε we have inf x ∈ R d P { Λ ℓ ( x ) is s -SEL for H ε,ω } ≥ − e − ℓ ζ . (4.109) igensystem Bootstrap Multiscale Analysis Then, if ℓ is sufficiently large, for L = ℓ γ we have inf x ∈ R d P { Λ L ( x ) is m -localizing for H ε,ω } ≥ − e − L ζ , (4.110) where m ≥ L − ( − τ + − sγ ) . (4.111) Proof.
We let B N , S N and E N as in the proof of Lemma 4.7. We proceed as inthe proof of Lemma 4.7. Using (4.109), (2.9) and the fact that events on disjointboxes are independent, we have P {B c } ≤ (cid:0) Lℓ (cid:1) ( N +1) d e − ( N +1) ℓ ζ = 2 ( N +1) d ℓ ( γ − N +1) d e − ( N +1) ℓ ζ (4.112) < e − ℓ γζ = e − L ζ , if N + 1 > ℓ ( γ − ζ and ℓ is sufficiently large. For this reason we take N = N ℓ = j ℓ ( γ − e ζ k = ⇒ P {B cN ℓ } ≤ e − L ζ for all ℓ sufficiently large . (4.113)Also, using (4.86) and (4.19), we have, P {S cN } ≤ Y ε (cid:16) L + 1) d N ℓ !( d d ) N | ℓ − (cid:17) ( L + 1) d e − (2 α − L β < e − L ζ (4.114)for sufficiently large L , since ( γ − e ζ < ( γ − β < γβ and ζ < β . Combining(4.112) and (4.114), we conclude that P {E N } > − e − L ζ . (4.115)To finish the proof we need to show that for all ω ∈ E N the box Λ L is m -localizing for H ε,ω , where m is given in (4.111). Following the proof of Lemma 4.7,we get σ ( H Λ L ) = σ G ( H Λ L ) ∪ σ B ( H Λ L ) and obtain an eigensystem { ( ψ x , λ x ) } x ∈ Λ L for H Λ L . To finish the proof we need to show that { ( ψ x , λ x ) } x ∈ Λ L is an m -localized eigensystem for Λ L . We proceed as in the proof of Lemma 4.7. We fix x ∈ Λ L , and take y ∈ Λ L such that k y − x k ≥ L τ , we have n ( ℓ )( ℓ + 1) + R X r =1 diam Υ r + 2 ℓ ≥ L τ . (4.116)where n ( ℓ ) is the number of times we used (4.96). Thus, recalling N = ⌊ ℓ ( γ − e ζ ⌋ and using (4.16), we have n ( ℓ ) ≥ ℓ +1 ( L τ − ℓ ⌊ ℓ ( γ − e ζ ⌋ − ℓ ) ≥ ℓℓ +1 (cid:16) ℓ γτ − − ℓ ( γ − e ζ − (cid:17) ≥ ℓ γτ − . (4.117)for sufficiently large ℓ since ( γ − e ζ + 1 < γτ . It follows from (4.98), | ψ x ( y ) | ≤ (cid:16) e − c ℓ s (cid:17) ℓ γτ − (4.118) ≤ e − L − ( − τ + 1 − sγ ) k y − x k igensystem Bootstrap Multiscale Analysis ℓ .We conclude that { ( ψ x , λ x ) } x ∈ Λ L is an m -localized eigensystem for Λ L , where m is given in (4.111), so the box Λ L is m -localizing for H ε,ω . Proposition 4.9.
Fix ε > . There exists a finite scale L ( ε ) with the followingproperty: Suppose for some scale L ≥ L ( ε ) , < ε ≤ ε , and m ≥ L − κ , where < κ < τ − γβ , we have inf x ∈ R d P { Λ L ( x ) is m -localizing for H ε,ω } ≥ − e − L ζ . (4.119) Then, setting L k +1 = L γk for k = 0 , , . . . , we have inf x ∈ R d P { Λ L k ( x ) is m -localizing for H ε,ω } ≥ − e − L ζk for k = 0 , , . . . . (4.120) Moreover, we have inf x ∈ R d P { Λ L k ( x ) is m -localizing for H ε,ω } ≥ − e − L ξk for all L ≥ L γ . (4.121) Lemma 4.10.
Fix ε > . Suppose for some scale ℓ , < ε ≤ ε , and m ≥ ℓ − κ ,where < κ < τ − γβ , we have inf x ∈ R d P { Λ ℓ ( x ) is m -localizing for H ε,ω } ≥ − e − ℓ ζ . (4.122) Then, if ℓ is sufficiently large, for L = ℓ γ we have inf x ∈ R d P { Λ L ( x ) is M -localizing for H ε,ω } ≥ − e − L ζ , (4.123) where M ≥ m (cid:16) − C d,ε ℓ − min { − τ ,γτ − ( γ − e ζ − ,τ − γβ − κ } (cid:17) ≥ L κ . (4.124)Lemma (4.10) and Proposition (4.9) follow from [EK1, Lemma 4.5], [EK1,Proposition 4.3], and [EK1, Section 4.3]. (Note that in [EK1], they assume m ≥ m − for a fixed m − . However, all the results still hold when m ≥ ℓ − κ , < κ <τ − γβ . (See the Lemmas for ♯ being LOC in Sections 2.2 and 2.3.)) To prove Theorem 1.6, first we assume (1.18), which is the same as (4.1) withletting Y = 400, for some length scales. We apply Proposition 4.1, obtaining asequence of length scales satisfying (4.2). Therefore (4.50) is satisfied for somelength scales. Applying Proposition 4.3, we get a length scale satisfying (4.51).It follows that (4.59) is satisfied since 0 < − τ + γ < τ . We apply Proposi-tion 4.4, obtaining a sequence of length scales satisfying (4.60). Therefore, In view igensystem Bootstrap Multiscale Analysis Y = 400 − s . We apply Proposi-tion 4.6, obtaining a sequence of length scales satisfying (4.82). Therefore (4.109)is satisfied for some length scales. Applying Proposition 4.8, we get a length scalesatisfying (4.110). It follows that (4.119) is satisfied since 0 < − τ + − sγ < τ − γβ .We apply Proposition 4.9, getting (4.121), so (1.18) holds. Theorem 1.7 is an immediate consequence of Theorem 1.6 and Proposition 5.1.
Proposition 5.1.
Given q > dα and ε > , set θ ε,L = (cid:4) L (cid:5) log L log (cid:18) L − q dε (cid:19) . (5.1) Then inf x ∈ R d P { Λ L ( x ) is θ ε,L -polynomially localizing for H ε,ω } (5.2) ≥ − K ( L + 1) d (cid:0) dε + 2 L − q (cid:1) α . In particular, given θ > and P > , there exists a finite scale L ( q, θ, P ) suchthat for all L ≥ L ( q, θ, P ) and < ε ≤ d L − q we have inf x ∈ R d P { Λ L ( x ) is θ -polynomially localizing for H ε,ω } ≥ − P . (5.3)Proposition 5.1 shows that the starting hypothesis for the bootstrap multiscaleanalysis of Theorem 1.6 can be fulfilled .To prove Proposition 5.1, we will use the following lemma given in [EK1,Lemma 4.4]. Lemma 5.2 ([EK1, Lemma 4.4]) . Let H ε = − ε ∆ + V on ℓ ( Z d ) , where V is abounded potential and ε > . Let Θ ⊂ Z d , and suppose there is η > such that | V ( x ) − V ( y ) | ≥ η for all x, y ∈ Θ , x = y. (5.4) Then for ε < η d the operator H ε, Θ has an eigensystem { ( ψ x , λ x ) } x ∈ Θ such that | λ x − λ y | ≥ η − dε > for all x, y ∈ Θ , x = y, (5.5) and for all y ∈ Θ we have | ψ y ( x ) | ≤ (cid:16) dεη − dε (cid:17) | x − y | for all x ∈ Θ . (5.6) igensystem Bootstrap Multiscale Analysis Proof of Proposition 5.1.
Let ε > L = Λ L ( x ) for some x ∈ R d . Let η = 4 dε + L − q and suppose | V ( x ) − V ( y ) | ≥ η for all x, y ∈ Θ , x = y. (5.7)It follows from Lemma 5.2 that H ε, Λ L has an eigensystem { ( ψ x , λ x ) } x ∈ Λ L satisfying(5.5) and (5.6). We conclude from (5.5) that Λ L is polynomially level spacing for H ε . Moreover, using (5.6) and k x k ≤ | x | , for all y, x ∈ Λ L with k x − y k ≥ L ′ wehave | ψ y ( x ) | ≤ (cid:16) dεη − dε (cid:17) k x − y k = L − k x − y k log L log ( η − dε dε ) (5.8)= L − k x − y k log L log (cid:16) L − q dε (cid:17) ≤ L − θ ε,L with θ ε,L as in (5.1). Therefore Λ L ( x ) is θ -polynomially localizing.We have P { Λ L is not θ ε,L -polynomially localizing } ≤ P { (5.7) does not hold } (5.9) ≤ ( L +1) d S µ (cid:0) (cid:0) dε + L − q (cid:1)(cid:1) ≤ K ( L + 1) d (cid:0) dε + 2 L − q (cid:1) α , which yields (5.2). (We assumed 8 dε + 2 L − q ≤
1; if not (5.2) holds trivially.)If 0 < ε ≤ d L − q , for sufficiently large L we have θ ε,L ≥ θ , andinf x ∈ R d P { Λ L ( x ) is θ -polynomially localizing for H ε,ω } ≥ − P , (5.10)since αq − d > Acknowledgement.
A.K. wants to thank Alexander Elgart for many discus-sions.
References [Ai] Aizenman, M.: Localization at weak disorder: some elementary bounds.Rev. Math. Phys. , 1163-1182 (1994)[AiSFH] Aizenman, M., Schenker, J., Friedrich, R., Hundertmark, D.: Finitevolume fractional-moment criteria for Anderson localization. Commun.Math. Phys. , 219-253 (2001)[AiENSS] Aizenman, M., Elgart, A., Naboko, S., Schenker, J., Stolz, G.: Momentanalysis for localization in random Schr¨odinger operators. Inv. Math. , 343-413 (2006)[AiM] Aizenman, M., Molchanov, S.: Localization at large disorder and ex-treme energies: an elementary derivation. Commun. Math. Phys. ,245-278 (1993) igensystem Bootstrap Multiscale Analysis , 389-426 (2005)[BuDM] Burkard, R., Dell’Amico, M., Martello, S.: Assignment problems . Soci-ety for Industrial and Applied Mathematics (SIAM), Philadelphia, PA,2009.[CoH] Combes, J.M., Hislop, P.D.: Localization for some continuous, randomHamiltonians in d-dimension. J. Funct. Anal. , 149-180 (1994)[Dr] von Dreifus, H.:
On the effects of randomness in ferromagnetic modelsand Schr¨odinger operators . Ph.D. thesis, New York University (1987)[DrK] von Dreifus, H., Klein, A.: A new proof of localization in the Ander-son tight binding model. Commun. Math. Phys. , 285-299 (1989).http://projecteuclid.org/euclid.cmp/1104179145[EK1] Elgart, A., Klein, A.: An eigensystem approach to An-derson localization. J. Funct. Anal. , 3465-3512 (2016).doi:10.1016/j.jfa.2016.09.008[EK2] Elgart, A., Klein, A.: Eigensystem multiscale analysis for Andersonlocalization in energy intervals. Preprint, arXiv:1611.02650[FK] Figotin, A., Klein, A.: Localization of classical waves I:Acoustic waves. Commun. Math. Phys. , 439-482 (1996).http://projecteuclid.org/euclid.cmp/1104287356[FroS] Fr¨ohlich, J., Spencer, T.: Absence of diffusion with Anderson tightbinding model for large disorder or low energy. Commun. Math. Phys. , 151-184 (1983)[FroMSS] Fr¨ohlich, J., Martinelli, F., Scoppola, E., Spencer, T.: Constructiveproof of localization in the Anderson tight binding model. Commun.Math. Phys. , 21-46 (1985)[GK1] Germinet, F., Klein, A.: Bootstrap multiscale analysis and localiza-tion in random media. Commun. Math. Phys. , 415-448 (2001).doi:10.1007/s002200100518[GK2] Germinet, F., Klein, A.: A comprehensive proof of localization forcontinuous Anderson models with singular random potentials. J. Eur.Math. Soc. , 53-143 (2013). doi:10.4171/JEMS/356[Kl] Klein, A.: Multiscale analysis and localization of random operators. In Random Schr¨odinger Operators . Panoramas et Synth`eses , 121-159,Soci´et´e Math´ematique de France, Paris 2008[KlM] Klein. A., Molchanov, S.: Simplicity of eigenvalues in the Andersonmodel. J. Stat. Phys. , 95-99 (2006). doi:10.1007/s10955-005-8009-7 igensystem Bootstrap Multiscale Analysis51