A Multi-scale Analysis Proof of the Power-law Localization for Random Operators on ${\Z}^d$
aa r X i v : . [ m a t h . SP ] A p r A MULTI-SCALE ANALYSIS PROOF OF POWER-LAWLOCALIZATION FOR RANDOM OPERATORS ON Z d YUNFENG SHI
Abstract.
In this paper we give a multi-scale analysis proof of power-law localization for random operators on Z d for arbitrary d ≥
1. Our results applyto potentials with singular distributions. Introduction
Since the seminal work of Anderson [And58], the investigation of localizationfor noninteracting quasi-particles in random media has attracted great attention inphysics and mathematics community. In mathematics, the first rigorous proof of lo-calization for random operators was due to Goldsheid-Molchanov-Pastur [GMP77].They established the Anderson localization for a class of 1 D continuous randomSchr¨odinger operators. In higher dimensions, Fr¨ohlich-Spencer [FS83] proved, ei-ther at high disorder or low energies, the absence of diffusion for some randomSchr¨odinger operators by developing the celebrated multi-scale analysis (MSA)method. Based on MSA of [FS83], [FMSS85, DLS85, SW86] finally obtainedthe Anderson localization at either high disorder or low energies. We should re-mark that the method of [FS83] was simplified and extended by von Dreifus-Klein[vDK89] via introducing a scaling argument. The proof of [vDK89] allows the sin-gular distributed potentials not permitted in [DLS85, SW86]. Later, the argumentof [vDK89] was generalized by Klein [Kle93] to prove the Anderson localization forAnderson model with exponential long-range hopping.The improved MSA has many applications in different mathematical problems.We want to mention two of them in quasi-periodic setting. The first one is aboutthe KAM theory for Hamiltonian PDEs. As is well-known, the early KAM tech-niques can only tackle PDEs with simple normal frequencies. However, the multiplenormal frequencies appear naturally in PDEs, which become more serious in higherspace dimensions. Significantly, by combining a modified MSA with some non-linear methods (such as Nash-Moser iteration), Craig-Wayne [CW93] and Bour-gain (see [Bou05] and references therein) can handle PDEs with multiple normalfrequencies. This has already led to major progress in infinitely dimensional KAMtheory. Secondly, the MSA has been largely enhanced to study localization forquasi-periodic operators. In quasi-periodic setting the strong dependence of thequasi-periodic potential at different lattice sites makes the standard Wegner esti-mate become generally invalid. The Wegner estimate is of course crucial to performthe MSA. In addition, in true higher dimensions setting (higher lattice dimensionand multi-phase), the one-dimensional methods based on Lyapunov exponents and Date : April 17, 2020.
Key words and phrases.
Power-law localization, Multi-scale analysis, Random operators. dynamical arguments seem also not applicable. Remarkably, Bourgain and hiscoauthors [Bou02, BGS02, Bou07] established the large deviation theorem by de-veloping MSA together with a series of powerful tools, such as the subharmonicfunction estimate and semi-algebraic geometry arguments. For recent results alongthese lines, we refer to [BK19, JLS20, Shi19].Interestingly, techniques in quasi-periodic setting also have important applica-tions in random case. Bourgain [Bou09] proved localization for a random Fr¨ohlichmodel at low energies by applying the subharmonic function estimate. Jitomirskaya-Zhu [JZ19] gave a new proof of Anderson localization for 1 D Anderson model at arbitrary disorder via large deviation estimates and subharmonic arguments (seealso [BDF +
19] for another proof).An alternative method for the proof of localization (for random operators),known as the fractional moment method (FMM), was developed by Aizenman-Molchanov [AM93]. This method also has numerous applications in localizationproblems. Remarkably, the FMM was enhanced to prove (strong) dynamical local-ization [Aiz94, ASFH01, AW15]. In particular, if the conditional probability dis-tributions of potentials were absolutely continuous , Aizenman-Molchanov [AM93]proved the first sharp power-law localization for random operators with polynomial long-range hopping on Z d at either high disorder or extreme energies by combin-ing with the Simon-Wolff criterion [SW86]. We should remark that the Green’sfunction estimate in FMM of all the above works required a mild condition (suchas H¨older continuity) on the probability distribution of the potential. However, inorder to deduce localization using FMM, the absolute continuity of the measurewas required (at least without further improvement). For the study of polynomial long-range random operators, we also refer to [JM99].The aim of present paper is twofold:(1) First, there is simply no MSA proof of power-law localization for polynomial long-range hopping random operators, as far as I know. Moreover, our localizationresult applies directly to potentials with H¨older continuous distributions (includingmany singular distributions).(2) The second one is about the MSA itself. All the above MSA involve esti-mating the inverses (or Green’s functions) of matrices with exponentially (or sub-exponentially (Gevrey type) in [Shi19]) decaying off-diagonal elements. We areinterested in to what extent the MSA could be applied to matrices with polyno-mially (Sobolev type) decaying off-diagonal elements in localization problems. In( sub ) exponential case to perform MSA it needs to control the Green’s functionsby iterating the geometric resolvent identity . In general, this argument may not beextended to polynomial case. This motivates us to develop new methods initiatedby Kriecherbauer [Kri98] and Berti-Bolle [BB13] in dealing with quasi-periodic so-lutions for nonlinear PDEs. More precisely, we will directly construct left inverses and then obtain decaying estimate of Green’s functions.2. Main Results
Here is the set-up for our main results:
MSA PROOF OF POWER-LAW LOCALIZATION 3
Polynomial Long-range Hopping Random Operators.
Define on Z d the polynomial long-range hopping T ( m, n ) = ( | m − n | − r , for m = n with m, n ∈ Z d , , for m = n ∈ Z d , (2.1)where | n | = max ≤ i ≤ d | n i | and r > { V ω ( n ) } n ∈ Z d be independent identically distributed ( i.i.d. ) random variables(with common probability distribution µ ) on some probability space (Ω , F , P ) ( F a σ -algebra on Ω and P a probability measure on (Ω , F )).Throughout this paper we assume: • We have that d < r < ∞ . • The common distribution µ has compact support : For supp( µ ) = { x : µ ( x − ε, x + ε ) > ε > } ,supp( µ ) ⊂ [ − M, M ] , < M < ∞ . In this paper we study the dD random operators with polynomial long-rangehopping H ω = λ − T + V ω ( n ) δ nn ′ , λ ≥ , (2.2)where λ is the coupling constant describing the effect of disorder.Under above assumptions, H ω is a bounded self-adjoint operator on ℓ ( Z d ) foreach ω ∈ Ω. Denote by σ ( H ω ) the spectrum of H ω . A well-known result due toPastur [Pas80] can imply that there exists a set Σ ( compact and non-random ) suchthat for P almost all ω , σ ( H ω ) = Σ.2.2. Sobolev Norms of a Matrix.
Since we are dealing with matrices with poly-nomially decaying off-diagonal elements, the Sobolev norms introduced by Berti-Bolle [BB13] is useful.Fix s > d/ h k i = max { , | k |} with k ∈ Z d . Define for u = { u ( k ) } ∈ C Z d and s > k u k s = C ( s ) X k ∈ Z d | u ( k ) | h k i s , (2.3)where C ( s ) > s ≥ s ) k u u k s ≤ k u k s k u k s + C ( s ) k u k s k u k s with C ( s ) > C ( s ) = 1 / u u )( k ) = P k ′ ∈ Z d u ( k − k ′ ) u ( k ′ ). By Schur’s test and self-adjointness of T , we get (for r > d ) kT k ≤ sup m ∈ Z d X n = m | m − n | − r ≤ X n ∈ Z d \{ } | n | − r < ∞ , where k · k is the standard operator norm on ℓ ( Z d ). From [Kir08], we have for P almost all ω ,sup n ∈ Z d | V ω ( n ) | ≤ M. Thus we can assume sup n ∈ Z d | V ω ( n ) | ≤ M for all ω ∈ Ω. YUNFENG SHI
Let X , X ⊂ Z d be finite sets. Define M X X = {M = ( M ( k, k ′ ) ∈ C ) k ∈ X ,k ′ ∈ X } to be the set of all complex matrices with row indexes in X and column indexes in X . If Y ⊂ X , Y ⊂ X , we write M Y Y = ( M ( k, k ′ )) k ∈ Y ,k ′ ∈ Y for any M ∈ M X X . Definition 2.1.
Let
M ∈ M X X . Define for s ≥ s the Sobolev norms of M as: kMk s = C ( s ) X k ∈ X − X (cid:18) sup k ′ − k ′′ = k |M ( k ′ , k ′′ ) | (cid:19) h k i s , where C ( s ) > Remark 2.1.
From this definition, we have kT k r < ∞ for r < r − d/ Green’s Function Estimate.
The Green’s function plays a key role in spec-tral theroy. In this subsection we present the first main result about Green’s func-tion estimate.For n ∈ Z d and L >
0, define the cube Λ L ( n ) = { k ∈ Z d : | k − n | ≤ L } .Moreover, write Λ L = Λ L (0). The volume of a finite set Λ ⊂ Z d is defined to be | Λ | = | Λ L ( n ) | = (2 L + 1) d ( L ∈ N ) for example.If Λ ⊂ Z d , denote H Λ = R Λ H ω R Λ , where R Λ is the restriction operator. Definethe Green’s function (if it exists) as G Λ ( E ) = ( H Λ − E ) − , E ∈ R . Let us introduce good cubes in Z d . Definition 2.2.
Fix τ ′ > δ ∈ (0 ,
1) and d/ < s ≤ r < r − d/
2. We call Λ L ( n )is ( E, δ )- good if G Λ L ( n ) ( E ) exists and satisfies k G Λ L ( n ) ( E ) k s ≤ L τ ′ + δs for ∀ s ∈ [ s , r ] . Otherwise, we call Λ L ( n ) is ( E, δ )- bad . We call Λ L ( n ) an ( E, δ )- good (resp. ( E, δ )- bad ) L -cube if it is ( E, δ )- good (resp. ( E, δ )- bad ). Remark 2.2.
Let ζ ∈ ( δ,
1) and τ ′ − ( ζ − δ ) r <
0. Suppose that Λ L ( n ) is ( E, δ )- good . Then we have for L ≥ L ( ζ, τ ′ , δ, r , d ) > | n ′ − n ′′ | ≥ L/ | G Λ L ( n ) ( E )( n ′ , n ′′ ) | ≤ | n ′ − n ′′ | − (1 − ζ ) r . (2.4)Assume the following relations hold true: − (1 − δ ) r + τ ′ + 2 s < , − ξr + τ ′ + ατ + (3 + δ + 4 ξ ) s < ,α − (2 τ ′ + 2 ατ + (5 + 4 ξ + 2 δ ) s ) + s < τ ′ , (2.5)where α, τ, τ ′ , r > , ξ > , s > d/ δ ∈ (0 , x ] the integer part of x ∈ R . In what follows we let E be in aninterval I satisfying | I | ≤ I ∩ [ −kT k − M, kT k + M ] = ∅ . The main resultabout Green’s function estimate is: MSA PROOF OF POWER-LAW LOCALIZATION 5
Theorem 2.3.
Fix any J ∈ N and let L = [ l α ] ∈ N with l ∈ N . Suppose thatrelations (2.5) hold true. Then there exists l = l ( kT k r , M, J, α, τ, ξ, τ ′ , δ, r , s , d ) > such that if l ≥ l , ( P1 ) P ( ∃ E ∈ I s . t . Λ l ( m ) and Λ l ( n ) are ( E, δ ) − bad ) ≤ l − p for all | m − n | > l ,and ( P2 ) P (dist( σ ( H Λ L ( m ) ) , σ ( H Λ L ( n ) )) ≤ L − τ ) ≤ L − p / for all | m − n | > L ,then we have P ( ∃ E ∈ I s . t . Λ L ( m ) and Λ L ( n ) are ( E, ξα ) − bad ) ≤ C ( d ) L − J ( α − p − d ) + L − p / for all | m − n | > L . Remark 2.3. • In this theorem regular assumptions on µ are not needed. • If we assume further in this theorem (1 + ξ ) /α ≤ δ and p > αd + 2 αp/J ,then the “propagation of smallness” for probability occurs (see Theorem4.4 in the following for details).2.4. Power-law Localization.
A sufficient condition for the validity of ( P1 ) and( P2 ) in Theorem 2.3 can be derived from some regular assumption on µ .Let us recall the H¨older continuity of a distribution defined in [CKM87]. Definition 2.4 ([CKM87]) . We will say a probability measure µ is H¨older contin-uous of order ρ > K ρ ( µ ) = inf κ> sup < | a − b |≤ κ | a − b | − ρ µ ([ a, b ]) < ∞ . (2.6)In this case will call K ρ ( µ ) > µ . Remark 2.4. • Let µ be H¨older continuous of order ρ (i.e., K ρ ( µ ) > < κ < K ρ ( µ ), there is some κ = κ ( κ, µ ) > µ ([ a, b ]) ≤ κ − | a − b | ρ for 0 ≤ b − a ≤ κ . (2.7) • If µ is absolutely continuous with a density in L q with 1 < q ≤ ∞ , then µ is H¨older continuous of order 1 − /q , and K − /q ( µ ) ≥ k d µ d x k − L q , here d x means the Lesbesgue measure on R . • There are singular continuous µ which are H¨older continuous of some order ρ > power-law localization: Theorem 2.5.
Let H ω be defined by (2 . with the common distribution µ beingH¨older continuous of order ρ > , i.e., K ρ ( µ ) > . Let r ≥ max { d +23 ρdρ , d } .Fix any < κ < K ρ ( µ ) . Then there exists λ = λ ( κ, µ, ρ, M, r, d ) > such that for λ ≥ λ , H ω has pure point spectrum for P almost all ω . Moreover, for P almost all ω , there exists a complete system of eigenfunctions ψ ω = { ψ ω ( n ) } n ∈ Z d satisfying | ψ ω ( n ) | ≤ C ( ω ) h n − n ( ω ) i − r/ for some C ( ω ) > and n ( ω ) ∈ Z d . Remark 2.5.
YUNFENG SHI • One may replace max { d +23 ρdρ , d } with a smaller one. Actually, if µ is absolutely continuous , it has been proven by Aizenman-Molchanov [AM93]that power-law localization holds for r > d using FMM. • In a forthcoming paper our new method will be applied to prove multi-particle power-law localization for quantum particles moving on Z d . • It may be a challenging problem to extend present result to polynomial long-range hopping Anderson-Bernoulli model (see [BK05] and [DS20] forthe tight-binding Anderson-Bernoulli models).2.5.
The Strategy of Proof.
Our proof is based essentially on Fr¨ohlich-Spencertype MSA [FS83]. In particular, we employ heavily the simplified MSA of vonDreifus-Klein [vDK89] (see also [Kir08] by Kirsch). However, one of the key ingre-dients in our proof is different from that in [FS83, vDK89] for which the geometricresolvent identity was iterated to obtain the exponentially decaying of off-diagonalelements of Green’s functions. Instead, we directly estimate the left inverses oftruncated matrix via information on small scales Green’s functions and a prior ℓ norm bound of the inverse itself. This idea was initiated by Kriecherbauer [Kri98]to deal with matrices with sub-exponentially decaying (even more general case) off-diagonal elements, and largely extended by Berti-Bolle [BB13] to study matriceswith polynomially decaying off-diagonal elements in dealing with Sobolev quasi-periodic solutions for nonlinear Schr¨odinger equations (see [BB12] for wave equa-tions). It should be pointed out that the Coupling Lemma (i.e., Proposition 4.1 in[BB13]) of Berti-Bolle can not be used directly here: To obtain a quantitative lowerbound on the decaying rate of long-range hopping, we have to carefully analyze the separation property of bad sites appeared in Coupling Lemma (see Remark 3.1 inthis paper for more details). In order to handle bad sites near boundary, we willmake use of an argument developed by Jitomirskaya, Liu and the author [JLS20] inquasi-periodic setting. Furthermore, our new Coupling Lemma (i.e., Lemma 3.2)permits separation distance of order l ξ ( ξ >
0) without increasing the diameterorder (of order also l ξ ) of bad sites clusters. Once the Green’s function estimatewas established, the proof of localization can be be accomplished with a polynomial long-range Shnol’s Theorem in [Han19].We think that our formulations in this paper may have applications in localiza-tion problems for other models.The structure of the paper is as follows. The proof of Theorem 2.3 is given in § §
4, verification of the assumptions ( P1 ) and ( P2 ) in Theorem 2.3 is presented.Moreover, the whole MSA on Green’s functions is also proved there. In §
5, the proofof Theorem 2.5 is complete. Some useful estimates are included in the appendix.3.
Proof of Theorem 2.3
Proof of Theorem 2.3.
The proof consists of a deterministic and a probabilisticpart.We begin with the following definition.
Definition 3.1.
We call a site n ∈ Λ ⊂ Z d is ( l, E, δ )- good with respect to ( w.r.t )Λ if there exists some Λ l ( m ) ⊂ Λ such that Λ l ( m ) is ( E, δ )- good and n ∈ Λ l ( m )with dist( n, Λ \ Λ l ( m )) ≥ l/
2. Otherwise, we call n ∈ Λ ⊂ Z d is ( l, E, δ )- bad w.r.t Λ. MSA PROOF OF POWER-LAW LOCALIZATION 7
We then prove a key
Coupling Lemma . Recall that E ∈ I with | I | ≤ I ∩ σ ( H ω ) = ∅ . Lemma 3.2 ( Coupling Lemma ) . Let L = [ l α ] . Assume the following: • The relations (2.5) hold true. • We can decompose Λ L ( n ) into two disjoint subsets Λ L ( n ) = B ∪ G with thefollowing properties: We have B = ∪ ≤ j ≤ J Ω j , where for each j , diam(Ω j ) ≤ C ⋆ l ξ ( C ⋆ > ), and for j = j ′ , dist(Ω j , Ω j ′ ) ≥ l ξ . For any k ∈ G , k is ( l, E, δ ) - good w.r.t Λ L ( n ) . • k G Λ L ( n ) ( E ) k ≤ L τ . Then for l ≥ l ( kT k r , M, C ⋆ , α, τ, ξ, τ ′ , δ, r , s , d ) > , we have Λ L ( n ) is ( E, ξα ) - good . Remark 3.1.
The main scheme of the proof is definitely from Berti-Bolle [BB13]in dealing with nonlinear PDEs. We provide the main changes and improvementsneeded in the present setting: • First, the definition of good (or bad ) sites in the present is different fromthat in [BB13]. In fact, our bad sites contain more elements. To over-come this problem, we apply techniques from [JLS20] to give geometric descriptions of bad sites. In particular, we can handle bad sites near theboundary of Λ L ( n ). • We can tackle separation distance of order l ξ without increasing the diam-eter order (of order also l ξ ) of bad sites clusters. This improvement is im-portant to obtain quantitative bound (i.e., r ≥ max { (100 /ρ + 23) d, d } ). • Our result allows any decay rate of the Green’s functions (i.e., any δ ∈ (0 , δ ∈ (0 , /
2) as in [BB13].
Proof.
The Sobolev norms introduced in [BB13] is convenient to the proof. Below,we collect some useful properties of Sobolev norms for matrices (see [BB13] fordetails): • ( Interpolation property ): Let
B, C, D be finite subsets of Z d and let M ∈ M CD , M ∈ M BC . Then for any s ≥ s , kM M k s ≤ (1 / kM k s kM k s + ( C ( s ) / kM k s kM k s , (3.1)and kM M k s ≤ kM k s kM k s , (3.2) kM M k s ≤ C ( s ) kM k s kM k s , (3.3)where C ( s ) ≥ C ( s ) = 1. In particular, if M ∈ M BB and n ≥
1, then kM n k s ≤ kMk ns , kMk ≤ kMk s , (3.4) kM n k s ≤ C ( s ) kMk n − s kMk s . (3.5) YUNFENG SHI • ( Smoothing property ): Let
M ∈ M BC . Then for s ≥ s ′ ≥ M ( k, k ′ ) = 0 for | k − k ′ | < N ⇒ kMk s ≤ N − ( s − s ′ ) kMk s ′ , (3.6)and for N ≥ N ( s , d ) > M ( k, k ′ ) = 0 for | k − k ′ | > N ⇒ ( kMk s ′ ≤ N s ′ − s kMk s , kMk s ≤ N s + s kMk . (3.7) • ( Columns estimate ): Let
M ∈ M BC . Then for s ≥ kMk s ≤ C ( s , d ) max k ∈ C kM { k } k s + s , (3.8)where M { k } = ( M ( k , k )) k ∈ B,k = k ∈ M B { k } is a column sub-matrix of M . • ( Perturbation argument ): If
M ∈ M BC has a left inverse N ∈ M CB (i.e., N M = I , where I the identity matrix), then for all P ∈ M BC with kPk s kN k s ≤ / , the matrix M + P has a left inverse N P that satisfies kN P k s ≤ kN k s , (3.9) kN P k s ≤ C ( s )( kN k s + kN k s kPk s ) for s ≥ s . (3.10)Moreover, if kPk · kN k ≤ / , then kN P k ≤ kN k . (3.11)We then turn to the proof of Coupling Lemma .Write A = H X − E with X = Λ L ( n ). Let T X = R X T R X . For u ∈ C X with X = B ∪ G , define u G = R G u ∈ C G and u B = R B u ∈ C B . Consider the linearequation on C X : A u = h. (3.12)Following [BB13], we have three steps: Step 1: Reduction on good sitesLemma 3.3.
Let l ≥ l ( τ ′ , δ, r , s , d ) > . Then there exist M ∈ M XG and N ∈ M BG satisfying the following: kMk s ≤ C ( s , d ) l τ ′ +(1+ δ ) s , kN k s ≤ C ( r , s , d ) kT X k r l − (1 − δ ) r + τ ′ +2 s ≤ / , (3.13) and for all s > s : kMk s ≤ C ( s, s , d ) l τ ′ +(1+2 δ ) s ( l s kT X k s + kT X k s + s ) , (3.14) kN k s ≤ C ( s, s , d ) l τ ′ + δs ( l s kT X k s + kT X k s + s ) , (3.15) such that u G = N u B + M h. (3.16) Proof.
Fix k ∈ G . Then there exists some l -cube F k = Λ l ( k ) such that k ∈ F k ,dist( k, X \ F k ) ≥ l/ F k is ( E, δ )- good . Define Q k = λ − G F k ( E ) T X \ F k F k ∈ M X \ F k F k . Since F k is ( E, δ )- good and using Interpolation property (3.3), weobtain (for λ ≥ kQ k k r ≤ C ( r ) k G F k ( E ) k r kT X k r ≤ C ( r ) kT X k r l τ ′ + δr . (3.17) MSA PROOF OF POWER-LAW LOCALIZATION 9 By Interpolation property (3.1) and
Smoothing property (3.7), we have for s ≥ s (for | k − k ′ | > l , G F k ( E )( k ′ , k ) = 0), kQ k k s + s ≤ C ( s )( k G F k ( E ) k s + s kT X k s + k G F k ( E ) k s kT X k s + s ) ≤ C ( s )((2 l ) s k G F k ( E ) k s kT X k s + l τ ′ + δs kT X k s + s ) ≤ C ( s, d ) l τ ′ + δs ( l s kT X k s + kT X k s + s ) . (3.18)We now vary k ∈ G . Define the following operators:Γ( k ′ , k ) = ( , for k ′ ∈ F k , Q k ( k ′ , k ) , for k ′ ∈ X \ F k , and L ( k ′ , k ) = ( G F k ( E )( k ′ , k ) , for k ′ ∈ F k , , for k ′ ∈ X \ F k . From (3.12), we have u G + Γ u = L h (3.19)We estimate Γ ∈ M XG . Fix k ∈ G . By definition and dist( k, X \ F k ) ≥ l/ k ′ ∈ X \ F k and | k ′ − k | < l/
2, then k ′ ∈ F k . This implies Γ { k } ( k ′ , k ) = 0 for | k ′ − k | < l/
2. By
Columns estimate (3.8) and (3.17) and
Smoothing property (3.6), we obtain k Γ k s ≤ C ( s , d ) sup k ∈ G k Γ { k } k s ≤ C ( s , d ) sup k ∈ G ( l/ − r +2 s k Γ { k } k r ≤ C ( s , d ) sup k ∈ G ( l/ − r +2 s kQ k k r ≤ C ( r , s , d ) kT X k r l − (1 − δ ) r + τ ′ +2 s . (3.20)Similarly, for s ≥ s , we obtain by recalling (3.18) k Γ k s ≤ C ( s , d ) sup k ∈ G k Γ { k } k s + s ≤ C ( s , d ) sup k ∈ G kQ k k s + s ≤ C ( s, s , d ) l τ ′ + δs ( l s kT X k s + kT X k s + s ) . (3.21)We then estimate L ∈ M XG . Fix k ∈ G . By definition of L , if | k ′ − k | > l , then k ′ / ∈ F k . This implies L { k } ( k ′ , k ) = 0 for | k ′ − k | > l . By Columns estimate (3.8) and
Smoothing property (3.7), we have for s ≥ kLk s + s ≤ C ( s , d ) sup k ∈ G kL { k } k s +2 s ≤ C ( s , d ) sup k ∈ G (2 l ) s + s kL { k } k s ≤ C ( s, s , d ) sup k ∈ G l s + s k G F k ( E ) k s ≤ C ( s, s , d ) l s + τ ′ +(1+ δ ) s . (3.22)Note that we have − (1 − δ ) r + τ ′ + 2 s <
0. Thus for l ≥ l ( τ ′ , δ, r , s , d ) > , we have by (3.20) k Γ k s ≤ /
2. Recalling the
Perturbation argument (3.9) and(3.10), I + Γ G is invertible and satisfies k ( I + Γ G ) − k s ≤ , (3.23) k ( I + Γ G ) − k s ≤ C ( s ) k Γ k s for s ≥ s . (3.24)From (3.19), we have u G = − ( I + Γ G ) − Γ B u B + ( I + Γ G ) − L h and then N = − ( I + Γ G ) − Γ B , M = ( I + Γ G ) − L . Recalling
Interpolation property (3.1), (3.23) and (3.24), we have kN k s ≤ k ( I + Γ G ) − k s k Γ k s ≤ C ( r , s , d ) kT X k r l − (1 − δ ) r + τ ′ +2 s , kMk s ≤ k ( I + Γ G ) − k s kLk s ≤ C ( s , d ) l τ ′ +(1+ δ ) s , and for s ≥ s , kN k s ≤ C ( s )( k ( I + Γ G ) − k s k Γ k s + k ( I + Γ G ) − k s k Γ k s ) ≤ C ( s ) k Γ k s ≤ C ( s, s , d ) l τ ′ + δs ( l s kT X k s + kT X k s + s ) (by (3.21)) , kMk s ≤ C ( s )( k ( I + Γ G ) − k s kLk s + k ( I + Γ G ) − k s kLk s ) ≤ C ( s )( k Γ k s kLk s + k Γ k s kLk s ) ≤ C ( s, s , d ) l τ ′ +(1+2 δ ) s ( l s kT X k s + kT X k s + s ) (by (3.20) , (3.21) and (3.22)) . (cid:3) Step 2: Reduction on bad sitesLemma 3.4.
Let l ≥ l ( τ ′ , δ, r , s , d ) > . We have A ′ u B = Z h (3.25) where A ′ = A B + A G N ∈ M BX , Z = I − A G M ∈ M XX satisfy for s ≥ s , kA ′ k s ≤ C ( M )(1 + kT X k s ) , (3.26) kZk s ≤ C ( M, s , d )(1 + kT X k s ) l τ ′ +(1+ δ ) s , (3.27) kA ′ k s ≤ C ( M, s, s , d )(1 + kT X k s ) l τ ′ + δs ( l s kT X k s + kT X k s + s ) , (3.28) kZk s ≤ C ( M, s, s , d )(1 + kT X k s ) l τ ′ +(1+2 δ ) s ( l s kT X k s + kT X k s + s ) . (3.29) Moreover, ( A − ) B is a left inverse of A ′ .Proof. Since I ∩ [ −kT k − M, kT k + M ] = ∅ , sup ω,n | V ω ( n ) | ≤ M , | I | ≤ λ ≥ E ∈ I and n ∈ Z d , | V ω ( n ) − E | ≤ kT k + 2 M + 1 . MSA PROOF OF POWER-LAW LOCALIZATION 11
Thus for any s ≥ kAk s = k H X − E k s ≤ k λ − T X k s + kT k + 2 M + 1 ≤ kT X k s + kT k + M ) . (3.30)From (3.13), (3.30), Interpolation property (3.1) and (3.2), we have kA ′ k s ≤ kAk s + kAk s kN k s ≤ C ( M )(1 + kT X k s ) (for kT X k ≤ kT X k s ) , kZk s ≤ kAk s kMk s ≤ C ( M, s , d )(1 + kT X k s ) l τ ′ +(1+ δ ) s , and for s ≥ s , kA ′ k s ≤ kAk s + C ( s )( kAk s kN k s + kAk s kN k s ) ≤ C ( M, s, s , d )(1 + kT X k s ) l τ ′ + δs ( l s kT X k s + kT X k s + s ) , kZk s ≤ C ( s )( kAk s kMk s + kAk s kMk s ) ≤ C ( M, s, s , d )(1 + kT X k s ) l τ ′ +(1+2 δ ) s ( l s kT X k s + kT X k s + s ) . It is easy to see ( A − ) B is a left inverse of A ′ . (cid:3) Lemma 3.5 (Left inverse of A ′ ) . Let l ≥ l ( kT X k r , M, C ⋆ , τ, ξ, τ ′ , δ, r , s , d ) > .Then A ′ has a left inverse V satisfying for s ≥ d , kVk s ≤ C ( C ⋆ , s , d ) l ατ +(2+2 ξ ) s , (3.31) and for s > s , kVk s ≤ C ( M, C ⋆ , s, s , d )(1 + kT X k s ) l τ ′ +2 ατ +(4+4 ξ + δ ) s ( l (1+ ξ ) s kT X k s + kT X k s + s ) . (3.32) Proof.
The proof is based on perturbation of left inverses as in [BB13]. Let e Ω j bethe l ξ / j : e Ω j = { k : dist( k, Ω j ) ≤ l ξ / } . Let D ∈ M BX satisfy the following: D ( k, k ′ ) = A ′ ( k, k ′ ) , for ( k, k ′ ) ∈ [ j (Ω j × e Ω j ) , , for ( k, k ′ ) / ∈ [ j (Ω j × e Ω j ) . We claim that D has a left inverse W satisfying kWk ≤ L τ . Let | k − k ′ | < l ξ / R = A ′ − D . Since B = S j Ω j , we have k ∈ Ω j for some j , and then k ′ ∈ e Ω j ,which implies R ( k, k ′ ) = 0. Then recalling Smoothing property (3.1), we obtain kRk s ≤ ( l ξ / − r +2 s kRk r − s ≤ ( l ξ / − r +2 s kA ′ k r − s ≤ C ( M, r , s , d )(1 + kT X k r ) kT X k r l − ξr + τ ′ +(1+ δ +2 ξ ) s (by (3.28)) ≤ C ( M, r , s , d ) l − ξr + τ ′ +(1+ δ +2 ξ ) s . (3.33)Thus recalling (3.4) and the assumption kA − k ≤ L τ , kRk · k ( A − ) B k ≤ kRk s kA − k≤ C ( M, r , s , d ) l − ξr + τ ′ +(1+ δ +2 ξ ) s L τ ≤ C ( M, r , s , d ) l − ξr + τ ′ + ατ +(1+ δ +2 ξ ) s ≤ / L = [ l α ]) , where in the last inequality we use the fact that − ξr + τ ′ + ατ +(1+ δ +2 ξ ) s < l ≥ l ( M, α, τ, ξ, τ ′ , δ, r , s , d ) >
0. It follows from the
Perturbation argument (3.11) that D has a left inverse W satisfying kWk ≤ kA − k ≤ L τ .From [BB13], we know that W ( k, k ′ ) = W ( k, k ′ ) , for ( k, k ′ ) ∈ [ j (Ω j × e Ω j ) , , for ( k, k ′ ) / ∈ [ j (Ω j × e Ω j )is a left inverse of D . We then estimate kW k s . Since diam( e Ω j ) ≤ C ⋆ l ξ , wehave W ( k, k ′ ) = 0 if | k − k ′ | > C ⋆ l ξ . Using Smoothing property (3.7) yieldsfor s ≥ kW k s ≤ C ( C ⋆ , s, s , d ) l (1+ ξ )( s + s ) kWk ≤ C ( C ⋆ , s, s , d ) l (1+ ξ )( s + s )+ ατ . (3.34)Finally, recall that A ′ = D + R and W is a left inverse of D . We have by (3.33)and (3.34) kRk s kW k s ≤ C ( M, C ⋆ , r , s , d ) l − ξr + τ ′ + ατ +(3+ δ +4 ξ ) s ≤ / − ξr + τ ′ + ατ + (3 + δ + 4 ξ ) s < l ≥ l ( M, C ⋆ , α, τ, ξ, τ ′ , δ, r , s , d ) > Perturbation argument (3.9) and (3.10) again implies that A ′ has aleft inverse V satisfying kVk s ≤ kW k s ≤ C ( C ⋆ , s , d ) l ατ +(2+2 ξ ) s , kVk s ≤ C ( s )( kW k s + kW k s kRk s ) (by (3.10)) ≤ C ( C ⋆ , s, s , d ) l (1+ ξ )( s + s )+ ατ + C ( M, C ⋆ , s, s , d )(1 + kT X k s ) l τ ′ +2 ατ +(4+4 ξ + δ ) s ( l s kT X k s + kT X k s + s ) ≤ C ( M, C ⋆ , s, s , d )(1 + kT X k s ) l τ ′ +2 ατ +(4+4 ξ + δ ) s ( l (1+ ξ ) s kT X k s + kT X k s + s ) . (cid:3) Step 3: Completion of proof
Combining (3.12), (3.16) and (3.25) implies u G = M h + N u B , u B = VZ h. Thus ( A − ) B = VZ , ( A − ) G = M + N ( A − ) B . Then for s ≥ s , we can obtain by using Interpolation property (3.1) and
Smoothing property (3.7) k ( A − ) B k s ≤ C ( s )( kVk s kZk s + kVk s kZk s ) ≤ C ( M, C ⋆ , s, s , d )(1 + kT X k s ) l τ ′ +2 ατ +(5+4 ξ +2 δ ) s ( l (1+ ξ ) s kT X k s + kT X k s + s )+ C ( M, C ⋆ , s, s , d )(1 + kT X k s ) l τ ′ + ατ +(3+2 δ +2 ξ ) s ( l s kT X k s + kT X k s + s )(by (3.29) and (3.32)) ≤ C ( M, C ⋆ , s, s , d )(1 + kT X k s ) l τ ′ +2 ατ +(5+4 ξ +2 δ ) s ( l (1+ ξ ) s kT X k s + kT X k s + s ) . MSA PROOF OF POWER-LAW LOCALIZATION 13
We obtain the similar bound for k ( A − ) G k s . Thus for any s ∈ [ s , r ], kA − k s ≤ k ( A − ) B k s + k ( A − ) G k s C ( M, C ⋆ , s, s , d )(1 + kT X k r ) l τ ′ +2 ατ +(5+4 ξ +2 δ ) s ( l (1+ ξ ) s kT X k s + (2 L ) s kT X k r ) ≤ C ( M, C ⋆ , r , s , d ) kT X k r L α − (2 τ ′ +2 ατ +(5+4 ξ +2 δ ) s )+ s + α − (1+ ξ ) s ≤ L τ ′ + ξα s , where in the last inequality we use the third inequality in (2.5) and L > l ≥ l ( kT k r , M, C ⋆ , α, τ, ξ, τ ′ , δ, r , s , d ) > . This finishes the proof of
Coupling Lemma . (cid:3) We then turn to the proof of Theorem 2.3.
STEP 1: Deterministic stepLemma 3.6.
Assume that any pairwise disjoint ( E, δ ) - bad l -cubes contained in Λ L ( n ) has number at most J − . Suppose that k G Λ L ( n ) ( E ) k ≤ L τ . Then for l ≥ l ( M, J, α, τ, ξ, τ ′ , δ, r , s , d ) > , Λ L ( n ) is ( E, ξα ) - good .Proof. The main point here is to obtain the separation property of ( l, E, δ )- bad sites w.r.t Λ L ( n ).Let { Λ jl } ≤ j ≤ J − be any pairwise disjoint ( E, δ )- bad l -cubes contained in Λ L ( n ).Let Z ⊂ { , · · · , J − } satisfy for any j ∈ Z , dist(Λ l , Λ jl ) ≤ l ξ . Obviously,1 ∈ Z . If Z = { , · · · , J − } , we stop the process. If Z = { , · · · , J − } , fixsome j ∈ { , · · · , J − } \ Z and define Z ⊂ { , · · · , J − } \ Z with the followingproperty: For any j ∈ Z , dist(Λ j l , Λ jl ) ≤ l ξ . From the construction, we havedist(Λ jl , Λ j ′ l ) > l ξ if j ∈ Z , j ′ ∈ Z . Repeating above process, we are able to geta partition of { , · · · , J − } = S J j =1 Z j so that J ≤ J − jl , Λ j ′ l ) > l ξ if j ∈ Z i , j ′ ∈ Z i ′ for any i = i ′ . We further define e Λ j = S i ∈ Z j Λ il . Then diam( e Λ j ) ≤| Z j | l + ( | Z j | − l ξ ≤ Jl ξ for 1 ≤ j ≤ J ≤ J − . Let I j ⊂ Λ L ( n ) be the smallest interval Q di =1 [ a i , b i ] ⊂ Z d containing e Λ j . Thismeans diam( I j ) = diam( e Λ j ). A similar argument used in [JLS20] can imply thefollowing: For each I j there exists a larger interval Ω j ⊂ Λ L ( n ) containing I j with diam(Ω j ) ≤ diam( I j ) + 20 l ≤ Jl ξ . Moreover, for each n ′ S j ≤ J Ω j ,there exists some ( E, δ )- good cube Λ l ( n ′′ ) ⊂ Λ L ( n ) such that n ′ ∈ Λ l ( n ′′ ) anddist( n ′ , Λ L ( n ) \ Λ l ( n ′′ )) ≥ l/
2. Clearly, for j = j ′ , dist(Ω j , Ω j ′ ) ≥ l ξ − l > l ξ if l ≥ l ( ξ ) > B = S J j =1 Ω j , G = Λ L ( n ) \ B and C ⋆ = 30 J . Λ l ( k ′ ) Separation property of bad l -cubes Λ L ( n )Ω i Ω i Ω i Λ l ( k ) (cid:3) STEP 2: Probabilistic step
Fix m, n with | m − n | > L and write Λ = Λ L ( m ) , Λ = Λ L ( n ). We define thefollowing events for i = 1 , A i : Λ i is ( E, ξα ) − bad ,B i : either G Λ i ( E ) does not exist or k G Λ i ( E ) k ≥ L τ ,C i : Λ i contains J pairwise disjoint ( E, δ ) − bad l − cubes ,D : ∃ E ∈ I so that Λ and Λ are ( E, ξα ) − bad . Using Lemma 3.6 yields P ( D ) ≤ P [ E ∈ I ( A ∩ A ) ! ≤ P [ E ∈ I (( B ∪ C ) ∩ ( B ∪ C )) ! ≤ P [ E ∈ I ( B ∩ B ) ! + P [ E ∈ I ( B ∩ C ) ! + P [ E ∈ I ( C ∩ B ) ! + P [ E ∈ I ( C ∩ C ) ! ≤ P [ E ∈ I ( B ∩ B ) ! + 3 P [ E ∈ I C ! . (3.35)It is easy to see since ( P1 ) P [ E ∈ I C ! ≤ C ( d ) L Jd ( l − p ) J/ ≤ C ( d ) L − J ( α − p − d ) . (3.36)We then estimate the first term in (3.35). By ( P2 ), we obtain P [ E ∈ I ( B ∩ B ) ! ≤ P (cid:0) dist( σ ( H Λ ) , σ ( H Λ )) ≤ L − τ (cid:1) ≤ L − p / . (3.37) MSA PROOF OF POWER-LAW LOCALIZATION 15
Combining (3.35), (3.36) and (3.37), we have P ( D ) ≤ C ( d ) L − J ( α − p − d ) + L − p / (cid:3) Validity of ( P1 ) and ( P2 ) In this section we shall show the validity of ( P1 ) and ( P2 ) in Theorem 2.3. As aconsequence, we prove a complete MSA argument on Green’s functions estimates. Theorem 4.1.
Let µ be H¨older continuous of order ρ > (i.e., K ρ ( µ ) > ). Fix < κ < K ρ ( µ ) , E ∈ R and τ ′ > ( p + d ) /ρ . Then there exists L = L ( κ, µ, ρ, τ ′ , p, r , s , d ) > such that the following holds: if L ≥ L , then there is some λ = λ ( L , κ, ρ, p, s , d ) > and η = η ( L , κ, ρ, p, d ) > so that for λ ≥ λ , we have P ( ∃ E ∈ [ E − η, E + η ] s . t . Λ L ( m ) and Λ L ( n ) are ( E, δ ) − bad ) ≤ L − p for all | m − n | > L . Remark 4.1.
We will see in the proof λ ∼ L ( p + d ) /ρ κ − /ρ and η ∼ L − ( p + d ) /ρ κ /ρ .In addition, λ and η are independent of E . Proof.
Define the event R n ( ε ) : | V ω ( k ) − E | ≤ ε for some k ∈ Λ L ( n ) , where ε ∈ (0 ,
1) will be specified below. Then by (2.7), we obtain for 2 ε ≤ κ = κ ( κ, µ ) > P ( R n ( ε )) ≤ (2 L + 1) d µ ([ E − ε, E + ε ]) ≤ ρ (2 L + 1) d κ − ε ρ ≤ L − p , (4.1)which shows we can set ε = 2 − − d/ρ κ /ρ L − ( p + d ) /ρ . In particular, (4.1) holds for L ≥ L ( κ, µ, ρ, p, d ) > ω / ∈ R n ( ε ). Then for all | E − E | ≤ ε/ k ∈ Λ L ( n ), we have | V ω ( k ) − E | ≥ | V ω ( k ) − E | − | E − E | ≥ ε/ , which shows we can set η = ε/
2. Moreover, for D = R Λ L ( n ) ( V ω − E ) R Λ L ( n ) , wehave by Definition 2.1 that kD − k s ≤ C ( d ) /ε for s ≥ s . Note that k λ − T D − k s ≤ C ( s , d ) λ − ε − ≤ / λ ≥ λ = 2 C ( s , d ) ε − . We assume λ ≥ λ . Then by Perturbation argument (i.e., (3.9) and (3.10)) and H Λ L ( n ) − E = R Λ L ( n ) λ − T R Λ L ( n ) + D , we have k G Λ L ( n ) ( E ) k s ≤ kD − k s ≤ C ( d ) ε − , and for s ≥ s , k G Λ L ( n ) ( E ) k s ≤ C ( s, d )( ε − + λ − ε − ) ≤ C ( s, s , d ) ε − (for λ ≥ λ ∼ ε − ) . We restrict s ≤ s ≤ r in the following. In order to show Λ L ( n ) is ( E, δ )- good ,it suffices to let C ( r , s , d ) ε − = C ( ρ, r , s , d ) κ − /ρ L ( p + d ) /ρ ≤ L τ ′ , (4.2)which indicates we can allow L ≥ L ( κ, µ, ρ, τ ′ , p, r , s , d ) > . We should remarkhere (4.2) makes sense because of τ ′ > ( p + d ) /ρ .Finally, for | m − n | > L and λ ≥ λ , we have by i.i.d. assumptions that P ( ∃ E ∈ [ E − η, E + η ] s . t . Λ L ( m ) and Λ L ( n ) are ( E, δ ) − bad ) ≤ P ( R m ( ε )) P ( R n ( ε )) ≤ L − p (by (4.1)) . (cid:3) We then turn to the verification of ( P2 ). This will follow from a Wegner typeestimate established by Carmona-Klein-Martinelli [CKM87]. Lemma 4.2.
Let µ be H¨older continuous of order ρ > (i.e., K ρ ( µ ) > ). Thenfor any < κ < K ρ ( µ ) , we can find κ = κ ( κ, µ ) > so that P (dist( E, σ ( H Λ L ( n ) )) ≤ ε ) ≤ κ − ρ (2 L + 1) d (1+ ρ ) ε ρ for all E ∈ R , n ∈ Z d and for all ε > , L > with ε (2 L + 1) d ≤ κ . Proof.
Note that the long-range term λ − T in our operator is non-random . Thenthe proof becomes similar to that in Schr¨odinger operator case by Carmona-Klein-Martinelli [CKM87]. We omit the details here. (cid:3) We can then verify ( P2 ) in Theorem 2.3: Theorem 4.3 ( Verification of ( P2 )) . Let µ be H¨older continuous of order ρ > (i.e., K ρ ( µ ) > ). Fix < κ < K ρ ( µ ) . Then For L ≥ L ( κ, µ, ρ, τ, p, d ) > and τ > (2 p + (2 + ρ ) d ) /ρ, (4.3) we have P (dist( σ ( H Λ L ( m ) ) , σ ( H Λ L ( n ) )) ≤ L − τ ) ≤ L − p / for all | m − n | > L .Proof. Apply Lemma 4.2 with ε = 2 L − τ . Then we have by i.i.d. assumptions,(4.3) and L ≥ L ( κ, µ, ρ, τ, p, d ) > P (dist( σ ( H Λ L ( m ) ) , σ ( H Λ L ( n ) ))) ≤ X E ∈ σ ( H Λ L ( m ) ) P (dist( E, σ ( H Λ L ( n ) )) ≤ L − τ ) ≤ κ − ρ (2 L + 1) d (2+ ρ ) L − ρτ ≤ L − p / . (cid:3) MSA PROOF OF POWER-LAW LOCALIZATION 17
Finally, we will finish the whole MSA.
Theorem 4.4.
Let µ be H¨older continuous of order ρ > (i.e., K ρ ( µ ) > ). Fix E ∈ R with | E | ≤ kT k + M ) , and assume relations (2.5) and (4.3) hold true.Assume further that (1 + ξ ) /α ≤ δ , and p > αd + 2 αp/J for J ∈ N . Then for < κ < K ρ ( µ ) , there exists L = L ( κ, µ, ρ, kT k r , M, J, α, τ, ξ, τ ′ , δ, p, r , s , d ) > such that the following holds: For L ≥ L , there is some λ = λ ( L , κ, ρ, p, s , d ) > and η = η ( L , κ, ρ, p, d ) > so that for λ ≥ λ and k ≥ , we have P ( ∃ E ∈ [ E − η, E + η ] s . t . Λ L k ( m ) and Λ L k ( n ) are ( E, δ ) − bad ) ≤ L − pk for all | m − n | > L k , where L k +1 = [ L αk ] and L ≥ L . Remark 4.2. • In this theorem we also have λ ∼ L ( p + d ) /ρ κ − /ρ and η ∼ L − ( p + d ) /ρ κ /ρ .Usually, to prove localization we can choose L ∼ L . The key point ofMSA scheme is that the largeness of disorder (i.e., λ ) depends only onthe initial scales. The later iteration steps do not increase λ further. Weobserve also that λ and η are free from E . • In order to apply Theorem 2.3, we restrict | E | ≤ kT k + M ) in thistheorem. Actually, we have σ ( H ω ) ⊂ [ −kT k − M, kT k + M ]. Proof.
Let L = L ( κ, µ, ρ, τ ′ , p, r , s , d ) > L = max { L , l } , (4.4)where l = l ( kT k r , M, J, α, τ, ξ, τ ′ , δ, r , s , d ) is given by Theorem 2.3.Then applying Theorem 4.1 with L ≥ L , λ = λ ( L , κ, ρ, p, s , d ) and η = η ( L , κ, ρ, p, d ) yields P ( ∃ E ∈ [ E − η, E + η ] s . t . Λ L ( m ) and Λ L ( n ) are ( E, δ ) − bad ) ≤ L − p for all | m − n | > L and λ ≥ λ .Let L k +1 = [ L αk ] and L ≥ L .Assume for some k ≥ P ( ∃ E ∈ [ E − η, E + η ] s . t . Λ L k ( m ) and Λ L k ( n ) are ( E, δ ) − bad ) ≤ L − pk for all | m − n | > L k . Obviously, we have by (4.4) that L k ≥ L ≥ L ≥ l > l = L k , L = L k +1 , I = [ E − η, E + η ]) andTheorem 4.3 yields P ( ∃ E ∈ [ E − η, E + η ] s . t . Λ L k +1 ( m ) and Λ L k +1 ( n ) are ( E, δ ) − bad ) ≤ L − pk +1 for all | m − n | > L k +1 .This finishes the whole MSA. (cid:3) Shnol’ s Theorem: Proof of Theorem 2.5
Recall the Poisson’s identity: Let ψ = { ψ ( n ) } ∈ C Z d satisfy H ω ψ = Eψ . Assumefurther G Λ ( E ) exists for some Λ ⊂ Z d . Then for any n ∈ Λ, we have ψ ( n ) = − X n ′ ∈ Λ ,n ′′ / ∈ Λ λ − G Λ ( E )( n, n ′ ) T ( n ′ , n ′′ ) ψ ( n ′′ ) . (5.1)We then introduce the Shnol’s Theorem of [Han19] in long-range operator case,which is useful to prove our localization. We begin with the following definition. Definition 5.1 ([Han19]) . Let ε >
0. An energy E is called an ε -generalizedeigenvalue if there exists some ψ ∈ C Z d satisfying ψ (0) = 1 , | ψ ( n ) | ≤ C (1+ | n | ) d/ ε and H ω ψ = Eψ . We call such ψ the ε -generalized eigenfunction.The Shnol’s Theorem for H ω is: Lemma 5.2 ([Han19]) . Let r − d > ε > and let E ε be the set of all ε -generalizedeigenvalues of H ω . Then we have E ε ⊂ σ ( H ω ) , E ε = σ ( H ω ) and ν ( σ ( H ω ) \ E ε ) = 0 ,where ν denotes some complete spectral measure of H ω , and E ε is the closure of E ε . From Shnol’s Theorem, to prove the localization it suffices to show every ε -generalized eigenfunction belongs to ℓ ( Z d ). In fact, we can even show every ε -generalized (with 0 < ε ≤ c ( d ) ≪
1) eigenfunction ψ decays as | ψ ( n ) | ≤ | n | − r/ for | n | ≫ L = L , λ , η and I = [ E − η, E + η ] in Theorem 4.4.Recalling Theorem 4.4, we have for λ ≥ λ , k ≥ P ( ∃ E ∈ I s . t . Λ L k ( m ) and Λ L k ( n ) are ( E, δ ) − bad ) ≤ L − pk (5.2)for all | m − n | > L k , where L k +1 = [ L αk ] and L ≫ power-law localization. Proof of Theorem 2.5.
We choose appropriate parameters satisfying (2.5), (4.3),(1 + ξ ) /α ≤ δ , and p > αd + 2 αp/J for J ∈ N . For this purpose, we can set bydirect calculation the following: α = 6 , δ = 1 / , ξ = 2 . Let 0 < ε ≪ J ⋆ = J ⋆ ( d, ε ) to be the smallesteven integer satisfying for p = 6 d + ε , p > d + J ⋆ p. As a consequence, we can set τ = (14 /ρ + 1) d + O ( ε + ε/ρ ) , s = d/ ε, τ ′ = (42 /ρ + 11 / d + O ( ε + ε/ρ ) . Recalling Remark 2.2, we set ζ = 19 /
20. Then for r ≥ (94 /ρ +13) d , we obtain τ ′ 2) (for ε ≪ L ( n ) is ( E, / good and L ≥ L ( r , d ) > k G Λ L ( n ) ( E ) k ≤ L (42 /ρ +23 / d + O ( ε + ε/ρ ) , (5.3) | G Λ L ( n ) ( E )( n ′ , n ′′ ) | ≤ | n ′ − n ′′ | − r / for | n ′ − n ′′ | ≥ L/ . (5.4)For any k ≥ 0, we define the set A k +1 = Λ L k +1 \ Λ L k and the event: E k : ∃ E ∈ I s . t . Λ L k and Λ L k ( n ) (for ∀ n ∈ A k +1 ) are ( E, / − bad . MSA PROOF OF POWER-LAW LOCALIZATION 19 Thus from p = 6 d + ε, α = 6 and (5.2), P ( E k ) ≤ (2 L k + 1) d L − d + ε ) k ≤ C ( d ) L − (6 d +2 ε ) k , X k ≥ P ( E k ) ≤ X k ≥ C ( d ) L − (6 d +2 ε ) k < ∞ . By Borel-Cantelli Lemma, we have P ( E k occurs infinitely often) = 0 . If we set Ω to be the event s.t. E k occurs only finitely often, then P (Ω ) = 1 . Let E ∈ I be an ε -generalized eigenvalue and ψ be its generalized eigenfunction,where 0 < ε ≪ ψ (0) = 1. Suppose nowthere exist infinitely many L k so that Λ L k are ( E, / good . Then from Poisson’sidentity (5.1), (5.3) and (5.4), we obtain for r ≥ (100 /ρ + 15) d ,1 = | ψ (0) | ≤ X n ′ ∈ Λ Lk ,n ′′ / ∈ Λ Lk C ( d ) | G Λ Lk ( E )(0 , n ′ ) | · | n ′ − n ′′ | − r (1 + | n ′′ | ) d/ ε ≤ (I) + (II) , where(I) = X | n ′ |≤ L k / , | n ′′ | >L k C ( ε , d ) L (42 /ρ +23 / d + O ( ε + ε/ρ ) k ( | n ′′ | / − (100 /ρ +15) d | n ′′ | d/ ε , (II) = X L k / ≤| n ′ |≤ L k , | n ′′ | >L k C ( d ) | n ′ | − (5 /ρ +3) d | n ′ − n ′′ | − (100 /ρ +15) d | n ′′ | d/ ε . For (I), we have by (A.1),(I) ≤ C ( ε , d ) L (42 /ρ +27 / d + O ( ε + ε/ρ ) k L − (100 /ρ +15 − / d/ O ( ε + ε/ρ + ε ) k ≤ C ( ε , d ) L − d/ρ + O ( ε + ε/ρ + ε ) k → L k → ∞ ) . For (II), we have also by (A.1),(II) ≤ C ( ε , d ) L dk X | n ′′ | >L k | n ′′ | − (5 /ρ +3 − / d + O ( ε ) ≤ C ( ε , d ) L − (5 ρ/ / d + O ( ε ) k → L k → ∞ ) . This implies that for any ε -generalized eigenvalue E , there exist only finitely many L k so that Λ L k are ( E, / good .In the following we fix ω ∈ Ω .From the above analysis, if r ≥ (100 /ρ + 15) d we have shown there exists k ( ω ) > k ≥ k all Λ L k ( n ) with n ∈ A k +1 is ( E, / good . Wedefine another set e A k +1 = Λ L k +1 / \ Λ L k . Obviously, e A k +1 ⊂ A k +1 . In thefollowing we will show for r ≥ max { (100 /ρ + 23) d, d } , ε, ε ≪ k ≥ k = k ( κ, µ, ρ, r, d, ω ) > | ψ ( n ) | ≤ | n | − r/ for n ∈ e A k +1 . (5.5)Once (5.5) was established for all k ≥ k , it follows from S k ≥ k e A k +1 = { n ∈ Z d : | n | ≥ L k } that | ψ ( n ) | ≤ | n | − r/ for all | n | ≥ L k . That has shown H ω exhibits power-law localization on I . In order to finish the proof of Theorem 2.5,it suffices to pave [ −kT k − M, kT k + M ] by intervals of length η .We set r = r + 8 d . In the following we try to prove (5.5). Note that ω ∈ Ω and n ∈ e A k +1 ⊂ A k +1 .We know Λ L k ( n ) ⊂ A k +1 is ( E, / good . Then recalling (5.1) again, we have | ψ ( n ) | ≤ X n ′ ∈ Λ Lk ( n ) ,n ′′ / ∈ Λ Lk ( n ) C ( d ) | G Λ Lk ( n ) ( E )( n, n ′ ) | · | n ′ − n ′′ | − r (1 + | n ′′ | ) d/ ε ≤ (III) + (IV) , where(III)= X | n ′ − n |≤ L k / , | n ′′ − n | >L k C ( d ) L (42 /ρ +23 / d + O ( ε + ε/ρ ) k ( | n ′′ − n | / − r (1 + L k +1 + | n ′′ − n | ) d/ ε , (IV) = X L k / ≤| n ′ − n |≤ L k , | n ′′ − n | >L k C ( d ) | n ′ − n | − r / | n ′ − n ′′ | − r (1 + L k +1 + | n ′′ − n | ) d/ ε . For (III), we have by (A.1),(III) ≤ C ( ε , r, d ) L d/ ε k +1 L (42 /ρ +27 / d + O ( ε + ε/ρ ) k X | n ′′ − n | >L k | n ′′ − n | − r + d/ ε ≤ C ( ε , r, d ) L (42 /ρ +39 / d + O ( ε + ε/ρ + ε ) k L ( − r +3 d/ ε ) / k ≤ C ( ε , r, d ) L − r/ /ρ +21 / d + O ( ε + ε/ρ + ε ) k . For (IV), we have also by (A.1),(IV) ≤ C ( ε , r , d ) L d/ ε k +1 L dk X | n ′′ − n | >L k | n ′′ − n | − r / d/ ε ≤ C ( ε , r , d ) L d +6 ε k L ( − r / d/ ε ) / k ≤ C ( ε , r , d ) L − r / d/ ε k . Combining the above estimates and noting r ≥ max { (100 /ρ + 23) d, d } , we have | ψ ( n ) | ≤ C ( ε , r , d ) L ( − r / d/ ε +7 ε ) / k ≤ | n | − r/ , where we use | L k | ≥ | n | / ≫ n ∈ e A k +1 , and ε, ε ≪ (cid:3) Acknowledgements I would like to thank Svetlana Jitomirskaya for reading the earlier versions ofthe paper and constructive suggestions. This work was supported by National Nat-ural Science Foundation of China grant 11901010 and China Postdoctoral ScienceFoundation grant 2018M641050. Appendix A.We introduce a useful lemma: Lemma A.1. Let L > with L ∈ N and Θ − d > . Then we have that X n ∈ Z d : | n |≥ L | n | − Θ ≤ C (Θ , d ) L − (Θ − d ) / , (A.1) where C (Θ , d ) > depends only on Θ , d . MSA PROOF OF POWER-LAW LOCALIZATION 21 Proof. 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A new proof of localization in the Anderson tight bindingmodel. Comm. Math. Phys. , 124(2):285–299, 1989.(Y. Shi) School of Mathematical Sciences, Peking University, Beijing 100871, China E-mail address ::