Perturbative diagonalisation for Maryland-type quasiperiodic operators with flat pieces
Ilya Kachkovskiy, Stanislav Krymski, Leonid Parnovski, Roman Shterenberg
aa r X i v : . [ m a t h . SP ] F e b PERTURBATIVE DIAGONALISATION FOR MARYLAND-TYPEQUASIPERIODIC OPERATORS WITH FLAT PIECES
ILYA KACHKOVSKIY, STANISLAV KRYMSKI, LEONID PARNOVSKI,AND ROMAN SHTERENBERG
Abstract.
We consider quasiperiodic operators on Z d with unbounded monotone sam-pling functions (“Maryland-type”), which are not required to be strictly monotone andare allowed to have flat segments. Under several geometric conditions on the frequen-cies, lengths of the segments, and their positions, we show that these operators enjoyAnderson localization at large disorder. Introduction
This paper can be considered as a direct continuation of the earlier publication [12].We consider quasiperiodic Schr¨odinger operators on ℓ ( Z d ):(1.1) ( H ( x ) ψ ) n = ε (∆ ψ ) n + f ( x + ω · n ) ψ n , where ω = ( ω , . . . , ω d ) is the frequency vector , and ∆ is the discrete Laplacian:(∆ ψ ) n = X m ∈ Z d : | m − n | =1 ψ m . We will consider the regime of large disorder which, after rescaling, corresponds to small ε >
0. The function f , which generates the quasiperiodic potential, is a non-decreasingfunction(1.2) f : ( − / , / → ( −∞ , + ∞ ) , f ( − / ±
0) = ∓∞ , and is extended into R \ ( Z + 1 /
2) by 1-periodicity. As usual for quasiperiodic operators,the numbers { , ω , . . . , ω d } are assumed to be linearly independent over Q .Potentials of the form (1.2) will be called Maryland-type , after the classical Marylandmodel with f ( x ) = tan( πx ) (we refer the reader to the following, certainly not exhaustive,list of related publications [1, 4, 5, 6, 7, 8, 9, 10, 11]). In [12, Section 6], we considereda case where f was not strictly monotone and was allowed to have a single flat piece ofsufficiently small length h ; in particular, | h | < min j | ω j | . In this situation, we can treatthis flat piece as a single isolated resonance. In the present paper, we consider severalmore elaborate situations where the resonance is not isolated but has, in some sense, finitemultiplicity.The main result of the paper is a general Theorem 5.2, where we establish some sufficientconditions under which the operator H admits Anderson localization. We also list severalparticular examples and refinements in Section 6. The simplest new example (see Theorem Date : February 8, 2021. ℓ ( Z ), consider the operator (1.1) with thefunction f satisfying f ( x ) ≡ E, x ∈ [ a, a + L ] ⊂ ( − / , / . Additionally, assume that ω is Diophantine (see Section 2.2 for the definition), and that[ a − ω, a + L +2 ω ] ⊂ ( − / , / f is Lipschitz monotone outside [ a, a + L ],with some regularity conditions similar to [12]. Assume also that L is not a rationalmultiple of ω . Then the operator (1.1) has Anderson localization for 0 < ε < ε ( ω, L ).We would also like to mention the simplest possible class of operators that are not cov-ered by our approach. Suppose that f is constant on an interval [ a, a + L ] ⊂ ( − / , / S := { n ∈ Z d : x + ω · n ∈ [ a, a + L ] + Z } has an unbounded connected component for some x (here we define connectedness on the Z d graph with nearest neighbour edges). For example, this will happen if 0 < ω < ω < L .Our methods cannot cover such operators. In fact, it seems possible that such models donot demonstrate Anderson localization around the energy E = f | [ a,a + L ] .Our approach, essentially, is based on assuming the opposite of the above: that is, thatall connected components of the set S are bounded and are sufficiently far away fromeach other. If we are able to surround each component by a layer of lattice points n suchthat f ( x + ω · n ) has good monotonicity properties, then, after a partial diagonalisation,their monotonicity will “propagate”, in a weaker form, into the interior of the set S .Afterwards, we can apply a modified version of the main result of [12] to finalize thediagonalisation, see Proposition 2.1. Acknowledgments.
The authors would like to dedicate this paper to the memory ofJean Bourgain.The research of LP was partially supported by EPSRC grants EP/J016829/1 andEP/P024793/1. RS was partially supported by NSF grant DMS–1814664. IK was par-tially supported by NSF grant DMS–1846114.2.
Preliminaries: regularity of f and convergence of perturbationseries While the functions f under consideration will have flat pieces, a necessary assumptionfor all proofs below would be the existence of sufficiently many pieces with a good controlof monotonicity. The corresponding regularity conditions and convergence results aresummarized in this section.It will be convenient to not exclude the case x ∈ / Z + ω · Z d , where exactly onevalue of the potential in the operator (1.1) becomes infinite: say, f ( x + ω · n ). In thiscase, the natural limiting object is the operator on ℓ ( Z d \ { n } ) obtained from (1.1) byenforcing the Dirichlet condition ψ ( n ) = 0. The results of [12], which we will be using,extend “continuously” into these values of x , see [12, Remark 4.14], with a reasonableinterpretation of infinities, if one adds an infinite eigenvalue with an eigenvector e n . Here, { e n : n ∈ Z d } is the standard basis in ℓ ( Z d ). We will also use the notation { e j : j =1 , , . . . , d } for the standard basis in R d . For a subset A ⊂ R d , it will be convenient touse the notation ℓ ( A ) for ℓ ( A ∩ Z d ). For example, ℓ ([0 , N ]) = ℓ { , , , . . . , N } . ONVERGENCE OF PERTURBATION THEORY 3 C reg -regularity. Similarly to [12], we will always assume the following:(f1) f : ( − / , / → R is continuous, non-decreasing, f ( − / −∞ , f (1 / −
0) = + ∞ , and f is extended by 1-periodicity into R \ ( Z + 1 / f satisfies (f1). Let C reg >
0, and x ∈ ( − / , / f is C reg -regularat x , if:(cr0) The pre-image f − (( f ( x ) − , f ( x )+2)) ∩ ( − / , /
2) is an open interval (denotedby ( a, b )), and f | ( a,b ) is a one-to-one map between ( a, b ) and ( f ( x ) − , f ( x ) + 2).(cr1) Let D min ( x ) := inf x ∈ ( a,b ) f ′ ( x ) ≥ f ′ does not exist, consider thesmallest of the derivative numbers). Then,(2.1) D min ( x ) ≤ f ′ ( x ) ≤ C reg D min ( x ) , ∀ x ∈ ( a, b ) . (cr2) Define ( a , b ) = f − ( f ( x ) − , f ( x ) + 1) ⊂ ( a, b ), and g ( x ) = 1 f ( x ) − f ( x ) , x ∈ ( b , a + 1) , extended by continuity to g ( ± /
2) = 0 (recall that we also assume f ( x +1) = f ( x ),so that the interval ( b , a + 1) is essentially ( − / , / \ ( a , b ) together with thepoint 1 / − / | g ′ ( x ) | ≤ C reg D min ( x ) , x ∈ ( b , a + 1) . For convenience, we will require D min ( x ) ≥ C reg -regularity.This condition can always be achieved by rescaling.2.2. The frequency vector.
The frequency vector ω ∈ R d is called Diophantine if thereexist C dio , τ dio > k n · ω k := dist( n · ω, Z ) ≥ C dio | n | − τ dio , ∀ n ∈ Z d \ { } . Without loss of generality, we will always assume 0 < ω < . . . < ω d < /
2. The set ofDiophantine vectors with the above property will be denoted by DC C dio ,τ dio , implying thatthe dependence on d will be clear from the context. We will only use this definition with τ dio > d + 1.2.3. Operators with convergent perturbation series.
In [12], it was shown that,if f is C reg -regular on ( − / , /
2) and ω is Diophantine, then the Rayleigh–Schr¨odingerperturbation series (see below) converges for sufficiently small ε >
0. However, one canalso apply the construction from [12] in the case where C reg and D min themselves dependon ε , under some additional restrictions on the off-diagonal terms of H ( x ). In [12, Section6], an example of such an operator was considered. In this section, we will describe aslightly more general class of operators for which the construction from the end of [12,Section 6] can be applied, virtually, without any changes. The main results of the presentpaper will be obtained by reducing various operators to the class described in this section.We will consider long range quasiperiodic operators with variable hopping terms. A quasiperiodic hopping matrix is, by definition, a matrix with elements of the followingform(2.3) Φ mn ( x ) = ϕ m − n ( x + ω · ( m + n ) / , m , n ∈ Z d , I. KACHKOVSKIY, S. KRYMSKII, L. PARNOVSKI, AND R. SHTERENBERG where ϕ m : R → C are Lipschitz 1-periodic functions, satisfying the self-adjointness con-dition: ϕ m = ϕ − m . Let also k ϕ k ε := max { sup k k ϕ k k ∞ , ε sup k k ϕ ′ k k ∞ } . Define
Range (Φ) to be the smallest number L ≥ nm ≡ | m − n | > L . Wewill only consider hopping matrices of finite range. Note that (2.3) can be reformulatedas the following covariance property:Φ m + a , n + a ( x ) = Φ mn ( x + a · ω ) , m , n , a ∈ Z d . Fix some R ∈ N , and suppose that Φ , Φ , . . . is a family of quasiperiodic hopping matriceswith Range (Φ k ) ≤ kR , defined by a family of functions ϕ m , ϕ m , . . . . The class of operatorswe would like to consider will be of the following form:(2.4) H = V + ε Φ + ε Φ + . . . , ≤ ε < , where ( V ( x ) ψ ) n = f ( x + n · ω ) ψ n . One can easily check that, assuming k ϕ k ∞ = sup j k ϕ j k ∞ = sup j, m k ϕ j m k ∞ < + ∞ , ≤ ε < , the part Φ = ε Φ + ε Φ + . . . defines a bounded operator on ℓ ( Z d ).The central object of [12] is the Rayleigh–Schr¨odinger perturbation series , which is aformal series of eigenvalues and eigenvectors(2.5) E = E + εE + ε E + . . . , (2.6) ψ = ψ + εψ + ε ψ + . . . where, in a small departure from the notation of [12], we assume(2.7) λ = f ( x + n · ω ) , ψ = e n , ψ j ⊥ ψ for j = 0 . Under the above assumptions, we consider the eigenvalue equation H ( x ) ψ = Eψ as equality of coefficients of two power series in the variable ε :( V + ε Φ + ε Φ + . . . )( ψ + εψ + ε ψ + . . . ) = ( E + εE + ε E + . . . )( ψ + εψ + ε ψ + . . . ) . Assuming that f is strictly monotone, the above system of equations has a unique solutionsatisfying (2.7).It will also be convenient to consider a graph Γ( x ), whose set of vertices is Z d , andthere is an edge between m and n if Φ mn ( x ) = 0. The length of that edge is the smallest j > j mn ( x ) = 0.We can now describe the class of operators with convergent perturbation series whichwill be used in this paper. Let I , . . . , I k ⊂ ( − / , /
2) be disjoint closed intervals, and µ , . . . , µ k >
0. We will only consider µ j ∈ N in applications, but the argument can beeasily extended for µ j ∈ [0 , + ∞ ). We will make the following assumptions. ONVERGENCE OF PERTURBATION THEORY 5 (conv1) f satisfies (f1) from the previous section. Additionally, f is C reg -regular on (0 , \ ( I ∪ . . . ∪ I k ) with D min ≥ c ε µ j ≤ f ′ ( x ) ≤ c on I j . Here, f is allowed to be non-differentiable at somevalues of x , but then the inequality is required for all derivative numbers. As aconsequence, f is strictly monotone on ( − / , / x + ω · n ∈ I j . Any edge of Γ( x ) that starts at n has length atleast µ j + 1.The proof of the following result can be done along the same lines as the argument in [12,Proof of Theorem 6.2]. Proposition 2.1.
Under the above assumptions (conv1) – (conv4) , there exists ε = ε ( C reg , { µ j } kj =1 , C dio , τ dio , k ϕ k ε , c , c ) > such that, for < ε < ε , the perturbation series for the operator (2.4) is convergent, andthe operator H ( x ) satisfies Anderson localization for all x ∈ R . Remark 2.2.
Here, we use an additional refinement of [12], by introducing k ϕ k ε instead ofconsidering k ϕ k ∞ and k ϕ ′ k ∞ separately. The reason is that differentiating ϕ is “cheaper”than differentiating the denominators, and the latter has already been accounted for in(conv2).One can also consider the perturbation series in finite volume. By “finite volume”, wemean the case when the operator is restricted to a finite box in Z d . The results of [12],as well as Proposition 2.1, do not extend directly into that case (in the language of [12],certain loops required for cancellation are forbidden). However, in all our applications,the volume will be fixed before choosing a small ε , in which case one can apply the regularperturbation theory of isolated eigenvalues: Proposition 2.3.
Let H = V + ε Φ be a self-adjoint operator on ℓ ([0 , N ] d ) : ( Hψ ) n = V n ψ n + ε X m Φ nm ψ m , with the diagonal part V (in other words, Φ nn ≡ ). Assume that all values V k are disjoint: δ = max m , k : m = k | V m − V k | > . Then the coefficients of the perturbation series (2.5) , (2.6) satisfy | E j | ≤ C ( N, d ) j k Φ k j δ j , k ψ j k ≤ C ( N, d ) j k Φ k j δ j . As a consequence, both series converge for small ε > . Corollary 2.4.
Suppose that f satisfies ( f , and { , ω , . . . , ω d } are rationally indepen-dent. For any M > and C reg > there exists ε = ε ( M, C reg , ω ) > , δ = δ ( M, C reg , ω ) such that the following is true. For any x ∈ R , any box B in Z d with | B | ≤ M , any ε with < ε < ε and any n ∈ B , such that f is C reg -regular at x + ω · n , the operator H restricted to the box B has a unique simple eigenvalue E = E ( x ) with | E ( x ) − f ( x + ω · n ) | ≤ εδ − k ϕ k ∞ . I. KACHKOVSKIY, S. KRYMSKII, L. PARNOVSKI, AND R. SHTERENBERG
The corresponding eigenvector ψ , k ψ k ℓ ( B ) = 1 , can be chosen to have | ψ ( x ) − e n | ≤ εδ − k ϕ k ∞ . Moreover, both E and ψ are Lipschitz continuous at x and satisfy similar bounds: | E ′ ( x ) − f ′ ( x + ω · n ) | ≤ εδ − k ϕ ′ k ∞ , | ψ ′ ( x ) | ≤ εδ − k ϕ ′ k ∞ . Proof.
Since the components of ω are rationally independent and f is C reg -regular at x + ω · n , we have f ( x + ω · n ) separated from the other diagonal entries by a constantthat only depends on ω , C reg , and the size of the box. Then the argument follows fromProposition 2.3. The differentiability follows from differentiating the perturbation seriesin the variable x which, in turn, follows from (cr2). (cid:3) Remark 2.5.
Exact same bounds on the derivatives also hold for the infinite volumeeigenvectors (in that case, δ = δ ( C dio , τ dio , d )). As a consequence, for most practicalpurposes involving at most one derivative, regular eigenvalues can be considered to besmall perturbations of the original diagonal entries, both in finite and infinite volumes. Ina finite volume, the constants depend on the size of the box, but can be made independentof the location of the box.The following is a simple consequence of rational independence. Proposition 2.6.
Suppose that f satisfies ( f . For any M > there exists W = W ( f, M, ω ) such that, for any box B with | B | ≤ M , all eigenvalues of H B ( x ) , except forat most one, are bounded by W + ε k ϕ k ∞ in absolute value.Proof. The only possibility for the operator to have a very large eigenvalue is for x + ω · n to be close to Z + 1 /
2. However, this implies that all the remaining values of x + ω · m ,where m = n is from the same box, are away from Z + 1 /
2. Ultimately, the dependenceon ω is only through C dio and τ dio . (cid:3) Some properties of Schr¨odinger eigenvectors
In this section, we will summarize some basic properties of the eigenvectors of discreteSchr¨odinger operators in bounded domains. Let C ⊂ Z d . Denote the Laplace operatoron ℓ ( C ) (with the Dirichlet boundary conditions on Z d \ C ) by(∆ C ψ )( n ) = X m ∈ C : | m − n | =1 ψ ( m ) . Definition 3.1.
Let
A, B ⊂ Z d . We will define n ∈ A ∪ B to be directly reachable from B using the following recurrent definition:(1) Any point n ∈ B is directly reachable in 0 steps.(2) Let n ∈ A ∪ B , m ∈ A , | m − n | = 1 and the set { m ′ ∈ Z d : | n − m ′ | ≤ , m ′ = m } is contained in A ∪ B and only consists of directly reachable points in at most k − m to be directly reachable in at most k steps. ONVERGENCE OF PERTURBATION THEORY 7
The above definition can be understood as follows. The eigenvalue equation(3.1) X m : | m − n | =1 ψ ( m ) = ( E − V ( n )) ψ ( n )allows us to determine the value ψ ( m ) from the values { ψ ( m ′ ) : | m ′ − n | ≤ , m ′ = m } , where n ∈ A ∪ B is some point with | n − m | = 1. A directly reachable point, therefore,is a point m ∈ A ∪ B such that ψ ( m ) can be determined from the values of ψ on B usinga finite sequence of such operations. Definition 3.2.
We will say that B ⊂ Z d satisfies direct unique continuation property (DUCP) for A ⊂ Z d if all points of A are directly reachable from B .We will need the following very coarse quantitative version of the unique continuation.Better modifications to this lemma can be made in particular situations. Lemma 3.3.
Assume that B ⊂ Z d satisfies DUCP for A ⊂ Z d , and any point of A isdirectly reachable from B in at most N steps, and | V n | ≤ W for all n ∈ A ∪ B for some W > . Assume that H A ∪ B ψ = Eψ , and k ψ k ℓ ( A ∪ B ) ≥ . Then there exists m ∈ B suchthat | ψ ( m ) | ≥ | A ∪ B | − (2 d + | E | + W ) − N . Proof.
By contradiction: each step we can only increase the value of the eigenfunction bya factor of 2 d + | E | + W . (cid:3) Corollary 3.4.
In Lemma . , replace ∆ by ε ∆ . Then the conclusion can be restated as | ψ ( m ) | ≥ | A ∪ B | − (2 d ε + ( | E | + W )) − N ε N . Remark 3.5.
In Lemma 3 .
3, one can generalize the Laplace operator in the followingway: the weight of any edge connecting two adjacent points of B can be replaced by anynumber between − ψ m , m ∈ B \ A , one would not have to divide by these weights. Remark 3.6.
For a subset A ⊂ Z d and k ∈ N , let ∂ k A := { n ∈ Z d \ A : dist( n , A ) ≤ k } . One can easily check that B := ∂ A satisfies DUCP for any bounded A . Now, supposethat A is a box. Then, in Corollary 3.4, one can take N = ⌈ diam( B ) ⌉ . Moreover, inCorollary 3.4, one can take W to be the second largest element of the set {| V n | : n ∈ A ∪ ∂ A } . In other words, for any m ∈ A , one can calculate ψ ( m ) from the values of ψ on B througha sequence of applications of (3.1) without using the value V ( n ) of the potential at anyparticular point n . Thus, the value of the potential at one point is allowed to be arbitrarylarge and yet excluded from consideration in the bound from 3.4.This argument will be particularly useful in combination with Proposition 2.6, since itstates that, while we may not be able to control the largest eigenvalue on a box of givensize, we can control all remaining eigenvalues in terms of ω , f , and the size of the box. I. KACHKOVSKIY, S. KRYMSKII, L. PARNOVSKI, AND R. SHTERENBERG
We will also need some information about the eigenfunction decay, which follows fromelementary perturbation theory.
Proposition 3.7.
Let A and H A be as above, and let A = A ⊔ . . . ⊔ A n be a disjointunion. Assume that | V m − V n | ≥ η, ∀ m ∈ A j , n ∈ A k , j = k. Then there exists an orthonormal basis { ψ m : m ∈ A } of eigenfunctions of H A such thatif m ∈ A j , then (3.2) | ψ m ( n ) | ≤ C ( A, { A j } ) η − dist( n ,A j ) . In other words, if the values of V are clustered, then the corresponding eigenfunctionsdecay exponentially away from the clusters. The proof immediately follows from thestandard perturbation theory [13]. The statement is only meaningful for η ≫
1, otherwiseone can absorb the bound into C ( A, { A j } ). The constant does not depend on V and on η ,as long as (3.2) is satisfied. In particular, it does not depend on the energy range inside acluster; additionally, any dependence on the number of lattice points in each cluster canalso be absorbed into C . Remark 3.8.
In Proposition 3.7, one can replace the Laplacian by a weighted Laplacianas long as the weights are bounded above by 1 in absolute value, with the same constants.One can also replace it by a long range operator if the distance function is modifiedaccordingly.The following statement, which also follows from perturbation theory for 2 × Proposition 3.9.
Under the assumptions of Proposition . , replace ∆ A by a weightedLaplacian, with all weights bounded by in absolute value, and the weights of all edgesconnecting points from different clusters are bounded by ε . Denote by H ′ A the operator H A with weights of all edges between different clusters replaced by zero. Then | λ j ( H ′ A ) − λ j ( H A ) | ≤ C ( A, { A j } ) ε η − , where λ j ( H ) denotes the j th eigenvalue of H in the increasing order.Proof. This is a standard argument from perturbation theory. For each pair of points n , m from different clusters, let P = h e n , ·i e n + h e m , ·i e m , and consider the operator H { m , n } := P H A P | ran P ;in other words, restrict H A to ℓ { e n , e m } . Since | V m − V n | ≥ η , for ε < η we havetwo eigenvectors of H { m , n } which are small perturbations of e m , e n , respectively. Denotethese (orthonormal) eigenvectors by f m , f n , and consider the unitary matrix with columns f m , f n : V = ( f m , f n ) . Let V := V ⊕ ℓ ( A \{ m , n } , ONVERGENCE OF PERTURBATION THEORY 9 and H A = V − H A V. The above procedure is called a partial × diagonalisation . One can check that, aftersuch transformation, the edge between m and n of length ε would disappear, and allother entries of the operator are changed at most by O ( ε η − ) (it is possible that theedge between m and n does not completely disappear, but its new length will be at most O ( ε η − )).Apply partial 2 × O ( ε η − ). Afterwards, and additional perturbation of the whole operator of size O ( ε η − )would completely eliminate those edges. Finally, one can restore the entries inside of eachcluster to their original form H ′ A by another perturbation of the order O ( ε η − ). Notethat the constants do depend on the size of the domain and geometry of clusters; however,they can be chosen uniformly in the magnitude of the potential (in fact, large values ofthe potential only make the situation easier), as long as the clusters are η -separated. (cid:3) We will also need the following property related to the differentiation of eigenvalueswith respect to a parameter. The following is often referred to as Hellmann–Feynmanvariational formula, see [13, Remark II.2.2]
Proposition 3.10.
Let t A ( t ) be a Lipschitz continuous family of self-adjoint matrices.Let λ ( t ) be a branch of an isolated eigenvalue of A ( t ) , and let ψ ( t ) be a corresponding ℓ -normalized eigenvector. Then (3.3) λ ′ ( t ) = ddt h A ( t ) ψ ( t ) , ψ ( t ) i = h A ′ ( t ) ψ ( t ) , ψ ( t ) i . Proof.
The remaining term in the right hand side of (3.3) is λ ( t ) ddt h ψ ( t ) , ψ ( t ) i , which isidentically zero due to normalization of ψ ( t ). (cid:3) In particular, if A ( t ) = A + f ( t ) h e j , ·i e j , then λ ′ ( t ) = f ′ ( t ) |h ψ ( t ) , e j i| . Proposition 3.11.
Under the assumptions of Proposition . , assume H ( t ) = H + f ( t ) h e k , ·i e k . Suppose that E = E ( t ) is an isolated eigenvalue of H ( t ) that is separated from the restof the spectrum by δ > . Assume that m ∈ A r , n ∈ A s . Denote by P ( t ) the spectralprojection onto the eigenspace associated with the eigenvalue E ( t ) . Then (cid:12)(cid:12)(cid:12)(cid:12) dP mn ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( A, { A j } ) | f ′ ( t ) | δ − η − dist( A r , { k } ) − dist( { k } ,A s ) . Proof.
The spectral theorem implies(3.4) |h ( H ( t ) − E ) − e m , e k i| ≤ Cδ − η − dist( A r , { k } ) . We also have ddt P ( t ) mn = (cid:18) ddt πi I | λ − E | = δ/ ( H A − λ ) − dλ (cid:19) mn , from which one can get the inequality by differentiating. (cid:3) Remark 3.12.
Our use of discrete unique continuation is relatively elementary. We referthe reader to [2, 3] for more advanced applications of unique continuation for discreteSchr¨odinger operators.4.
Matrix functions with Maryland-type diagonal entries
Let f : ( − / , / → R satisfy (f1) from Section 2. We will say that f is locallyLipschitz monotone at x ∈ ( − / , /
2) if f ′ ( x ) ≥ , ∀ x : | f ( x ) − f ( x ) | ≤ . This condition is a weaker version of C reg -regularity at x with D min ≥ f t ( x ) = f ( x ) + t { x − / } , where {·} denotes the fractional part. Assume that f, f t are extended into R by 1-periodicity. It is easy to see that f is locally Lipschitz monotone at all x ∈ ( − / , / f t is locally Lipschitz monotone at x , then all f t ′ , t ′ ≥ t , are also locallyLipschitz monotone at x .In the following theorem, we consider N × N matrix valued functions whose diagonalentries are locally Lipschitz monotone. Let { e , . . . , e N } denote the standard basis in C N . Theorem 4.1.
Let d , . . . , d N ∈ ( − / , / , min i = j | d i − d j | = δ > . Let also A t ( x ) = diag { f t ( x − d ) , . . . , f t ( x − d n ) } + ε Φ( x ) , A ( x ) = A ( x ) , where Φ( · ) is a continuous real symmetric -periodic N × N matrix-valued function on R .Assume that all eigenvalues of A t are simple for all t ∈ [0 , and all x ∈ R . Then thereexist c j = c j ( N ) > , j = 1 , such that, for ≤ ε k Φ k ∞ < c ( N ) δ, we have the following: (1) Suppose that f is locally Lipschitz monotone at x − d j . Then A ( x ) has a uniqueeigenvalue E j ( x ) and a unique ℓ -normalized real eigenvector ψ j ( x ) satisfying: (4.2) | E j ( x ) − f ( x − d j ) | ≤ c ( N ) ε k Φ k ∞ δ − , | ψ j ( x ) − e j | ≤ c ( N ) ε k Φ k ∞ δ − . (2) There exists a continuous -periodic real orthogonal matrix function U ( · ) : R → O ( N, R ) such that U − ( x ) A ( x ) U ( x ) is a diagonal matrix and j th column of U ( x ) coincides with ψ j ( x ) from (1) for all x where (1) is applicable. Remark 4.2.
One needs to specify the exact meaning of “simple” in the case when one ofthe matrix entries becomes infinite. It is easy to see that, as x approaches d j , exactly oneeigenvalue of A t ( x ) approaches ±∞ , and the remaining eigenvalues remain bounded. Infact, they approach the eigenvalues of A t with j th row and column removed, and thus canbe extended continuously through d j . Thus, we can require for all those finite eigenvaluesto be simple. Equivalently, we can require the distances between eigenvalues of A t ( x ) tobe bounded by some constant κ > t ∈ [0 ,
1] and all x ∈ R \ ( { d , . . . , d N } + Z ). ONVERGENCE OF PERTURBATION THEORY 11
Proof.
Part (1) follows from the standard perturbation theory of isolated eigenvalues: seealso Proposition 2.3. Note that, once we fix a real ℓ -normalized branch, it must be closeto e j or − e j , and requiring it to be close to e j completely determines this branch.Since f ( · ) is locally Lipschitz monotone on ( − / , / ψ j ( · ) : ( − / d j , / d j ) → R d , k ψ j ( x ) − e j k ≤ c ( N ) ε k Φ k ∞ δ − . Since the spectra of A t are simple, each ψ j has a unique real ℓ -normalized extensionin the parameter t into [0 , ψ j ( x, t ), and the correspondingeigenvalue branch by E j ( x, t ). For any x such that f is locally Lipschitz monotone at x − d j , any real eigenvector of A t ( x ) with eigenvalue E j ( x , t ) must be close to e j or − e j , uniformly in the parameter t . Since it is close to e j for t = 1, the same holds for all t . A similar argument can be applied to x near the endpoints. Since one (and only one)eigenvalue of A t ( x ) approaches infinity, the corresponding branch of eigenvector mustapproach e j or − e j as x → ± / d j , uniformly in t . Since for t = 1 it approaches + e j from both sides, the same holds for all t . In other words, it cannot suddenly change froma vector close to e j to a vector close to − e j , due to continuity in t and the fact that it hasto be close to e j or − e j for any particular t . (cid:3) Remark 4.3.
Under the assumptions of Theorem 4.1, one can take t → + ∞ in (4.1). Inthis case, ψ j ( x, t ) will be well defined for all t ≥ e j as t → ∞ .As a consequence, if the family { A t ( x ) } satisfies the assumptions of Theorem 4.1, thereis a unique family { U t ( x ) } of real orthogonal matrices such that U t ( x ) diagonalises A t ( x )and U t ( x ) → I as t → + ∞ . In this notation, we will have U ( x ) = U ( x ). One can usethis as an alternative definition of U ( x ). Proposition 4.4.
Fix
L > , a ∈ R . Let J be an N × N matrix satisfying the followingproperties: J ij = 0 for | i − j | > , and J ij = J ji ∈ R . Assume that J i,i +1 ≥ for allapplicable values of i , and that J ii ∈ [ a, a + L ] for all i . Then the spectrum of J is simple.Moreover, if E i = E j are two eigenvalues of J , then | E i − E j | ≥ c ( N, L ) > .Proof. The fact that the spectrum is simple is standard and follows from the fact thatthe any solution to the eigenvalue equation is uniquely determined from its value at onepoint. The rest follows from compactness. (cid:3)
Remark 4.5.
By rescaling, one can obtain a version of the proposition with | E i − E j | ≥ c ( N, εL ) ε provided that | J i,i +1 | > ε > Remark 4.6.
In Proposition 4.4, one can relax the requirement J ii ∈ [ a, a + L ] for theendpoint values J and J NN : one can remove this restriction for one of them, or for bothif one of them goes to + ∞ , and the other to −∞ . Remark 4.7.
In higher dimensions, Proposition 4.4 will be often used in combinationwith Proposition 3.9: if a cluster from the latter proposition has a linear shape (or issomewhat one-dimensional), then Proposition 4.4 guarantees that its eigenvalues will be cε -separated (after restricting the operator to the cluster). The presence of coupling termswith other clusters will shift the eigenvalues by at most O ( ε ), and will thus preserve theseparation condition. The moving block construction: general scheme
As a reminder, the standard basis of R d (and Z d ) is denoted by { e , e , . . . , e d } . Thetranslation operator on functions is defined by T n : ℓ ( Z d ) → ℓ ( Z d ) , ( T n ψ )( m ) = ψ ( m + n ) . Since we will use the following specific translations more often, we also denote T := T e , T ∗ := T − = T − e . Consider the operator (1.1) on ℓ ( Z d ) and assume that f satisfies (f1). Recall that,without loss of generality, we assumed0 < ω < ω < . . . < ω d < / . Fix some x ∈ R and C reg >
0. The construction below will depend on the choice of x ; however, any x would be equally applicable. A lattice point n ∈ Z d will be called regular for H ( x ) if f is C reg -regular at x + ω · n . As discussed in Corollary 2.4 and Remark2.5, for sufficiently small ε > H ( x ) has an eigenvalue close to f ( x + ω · n ) and thecorresponding eigenvector close to e n at any regular point n . The same also holds for arestriction of H onto any box containing n . The smallness of ε depends on the size ofthe box. Additionally, “close” in this case also applies to derivatives in x . A lattice pointwhich is not regular for H ( x ) will be, naturally, called singular for H ( x ). Let S sing ( x ) := { n ∈ Z d : n is singular for H ( x ) } . For the purposes of the construction, the point with infinite potential will be consideredregular. Since the components of ω are rationally independent, there is at most onesuch point. We will always assume that singular points are contained in a finite range ofenergies:(gen0) There is E reg > f is C reg -regular at any x with | f ( x ) | ≥ E reg .Let S ⊂ Z d be a finite subset possibly containing some singular points. Let also R ⊃ S be another finite subset, with the following properties:(gen1) S ∪ ( S + e ) ⊂ R ∩ ( R + e ).(gen2) For every x ∈ [ x , x + ω ], all points of ( R ∪ ( R + e )) \ ( S ∪ ( S + e )) are regularfor H ( x ).Let also(5.1) R − := R \ ( R + e ) , R + := ( R + e ) \ R, R := R ∩ ( R + e ) , R ′ := R ∪ ( R + e ) . We can represent R ′ as a disjoint union R ′ = R − ⊔ R + ⊔ R . Denote by H ′ R ′ ( x ) thefollowing family of operators on ℓ ( R ′ ): H ′ R ′ ( x ) := V R − ( x ) ⊕ H R ∪ R + ( x ); H ′ R ′ ( x + ω ) := H R − ∪ R ( x + ω ) ⊕ V R + ( x + ω ) , where V R ± ( x ) denotes just the potential part of H R ± ( x ), without the Laplacian, restrictedonto the domains R ± . For the values x ∈ [ x , x + ω ], we will linearly interpolate: H ′ R ′ ( x ) := (1 − s ) (cid:0) V R − ( x ) ⊕ H R ∪ R + ( x ) (cid:1) + s (cid:0) H R − ∪ R ( x + ω ) ⊕ V R + ( x + ω ) (cid:1) ,x = x + sω for s ∈ [0 , . Assume the following condition:
ONVERGENCE OF PERTURBATION THEORY 13 (gen3) Matrix families H ′ R ′ ( x ), H R ∪ R + ( x ), H R − ∪ R ( x ) satisfy the assumptions of Theorem4.1, for all x ∈ [ x , x + ω ].Technically, in order to apply Theorem 4.1, H ′ R ′ ( x ) has to be defined (possibly withinfinite entries) and be periodic for all x ∈ R . The exact form of the extension will notmatter. For the sake of clarity, assume that, for x ∈ [ x + ω , x + 1] the off-diagonal partsof H ′ R ′ ( x ) are linearly interpolated between those of H ′ R ′ ( x + ω ) and H ′ R ′ ( x ), so that H ′ R ′ ( x + 1) = H ′ R ′ ( x ), and then extended 1-periodically to R . The later construction(see the definition of the operator U mb0 ( x ) below) we will only use H ′ R ′ ( x ) for [ x , x + ω ].From the definitions, we have(5.2) H R ∪ R + ( x ) = T ∗ ( H R − ∪ R ( x + ω )) T, where T : ℓ ( R ∪ R + ) → ℓ ( R − ∪ R ) is the restriction of the translation map (which is abijection between R ∪ R + and R − ∪ R ). Denote by U R ′ ( x ), x ∈ [ x , x + ω ], the resultof applying Theorem 4.1 to the family H ′ R ′ ( x ). For x = x and x = x + ω , U R ′ ( x ) splitsinto a direct sum: U R ′ ( x ) = R − ⊕ U R ∪ R + ( x ); U R ′ ( x + ω ) = U R − ∪ R ( x + ω ) ⊕ R + . The matrices U R ∪ R + ( x ) and U R − ∪ R ( x + ω ) have real entries and, due to (5.2), diag-onalise the same operator, up to the identification of R − ∪ R with R ∪ R + . Therefore,they consist of the same eigenvectors, up to the ordering and the choice of the signs.On the other hand, (gen3) guarantees that both approach the identity as t → + ∞ , seeRemark 4.3. Therefore, they must be related in the same way as the operators (5.2) theydiagonalise:(5.3) U R ∪ R + ( x ) = T ∗ U R − ∪ R ( x + ω ) T. Extend U R ′ ( x ) into ℓ ( Z d \ R ′ ) by identity:(5.4) U mb0 ( x ) := U R ′ ( x ) ⊕ Z d \ R ′ . Then we have(5.5) T ∗ U mb0 ( x + ω ) T = U mb0 ( x ) , We start from U mb0 ( x ) defined for x ∈ [ x , x + ω ]. One can easily check that there is aunique extension of this family into x ∈ R satisfying(5.6) U mb0 ( x + ω ) = T − U mb0 ( x ) T, ∀ x ∈ R . This above construction will be called the diagonalisation of a moving block , which isreflected in the superscript mb. Note that the singular set also has the following covarianceproperty:(5.7) S sing ( x ) = S sing ( x − ω · n ) + n . As a consequence, if the above-mentioned set S from the moving block constructioncontains singular points for H ( x ), they will naturally correspond to singular points in S − e for H ( x + ω ). In the above construction, as x increases, the set S ‘moves’ along Z d with speed 1 /ω and traces a part of S sing , thus justifying the name of the construction.It will also be convenient to use the following language: a normal operator on ℓ ( Z d )is supported on R ′ if ℓ ( R ′ ) is its invariant subspace and the operator acts as an identity on ℓ ( Z d \ R ′ ). The above construction implies that U mb0 ( x ) is supported on R ′ for x ∈ [ x , x + ω ] and is supported on R ′ − m e for x ∈ [ x + mω , x + ( m + 1) ω ], where m ∈ Z .The operator family U mb0 ( x ) takes care only of a part of S sing ( x ) that corresponds to afinite block S that moves as x moves around R . We now need to cover the whole singularset by such blocks, and multiply the corresponding operators. At the same time, thefamily U mb0 ( x ) is not a quasiperiodic operator family: its entries are not even periodicin x . We will need to modify this family in order for it to be quasiperiodic. First, notethat the set S sing ( x ) is not only translationally covariant (5.7), but also translationallyinvariant: S sing ( x + 1) = S sing ( x ) . Therefore, with each moving block, we actually have a “train” of blocks, with a translationby 1. Thus, it is natural to consider(5.8) U mb1 ( x ) := Y m ∈ Z U mb0 ( x + m ) . We will later require (see (gen4) below) that the supports of U mb0 ( x + m ) are all disjoint.As a consequence, the above product will be well defined.Unlike U mb0 ( x ), the family U mb1 ( x ) is actually a periodic function in x , and is quasiperi-odic with respect to translations by ω :(5.9) U mb1 ( x ) = U mb1 ( x + 1) , U mb1 ( x + ω ) = T − U mb1 ( x ) T, ∀ x ∈ R . We now enforce the full covariance (quasiperiodicity) condition. Let ω ′ = ( ω , . . . , ω d ).For any n ′ ∈ Z d − \ { } , the family x T (0 , n ′ ) V ( x + n ′ · ω ′ ) T − (0 , n ′ ) describes a differentmoving block (in fact, a different train of moving blocks described above). Let(5.10) U mb2 ( x ) := Y n ′ ∈ Z d − T (0 , n ′ ) U ( x + n ′ · ω ′ ) T − (0 , n ′ ) . Each factor in (5.10) contains countably many copies of U mb0 ( x ). If well defined, U mb2 ( x )is a fully quasiperiodic operator family:(5.11) U mb2 ( x ) = U mb2 ( x + 1) , U mb2 ( x + ω · n ) = T − n U mb2 ( x ) T n , ∀ x ∈ R . If order for U mb1 and U mb2 to be well defined, we will require the last condition:(gen4) The supports of each copy of U mb0 inside of the operator (5.10) and (5.8) aredisjoint.The above construction will be referred to as a covariant family of moving blocks . Theset S will be called the base block . Remark 5.1.
Once we fix S and R ′ for x ∈ [ x , x + ω ], the position of other copies of S and R ′ , appearing in different factors of (5.10) and (5.8), is completely determined by S , R , and ω .One may need more than one such family to take care of all S sing ( x ), in which case wewill require all involved operators U mb2 to commute with each other (in other words, allsets R ′ under consideration will not overlap with each other). For simplicity, we will nowconsider the case with only one covariant moving block. Let H ( x ) := U mb2 ( x ) − H ( x ) U mb2 ( x ) . ONVERGENCE OF PERTURBATION THEORY 15
We will now outline the general strategy of the proof of localization. Our goal is to showthat, under some additional conditions on the sets R (which will impose some separationconditions on S ), the operator family H ( x ) will satisfy the assumptions of Proposition2.1. For the convenience of referencing, we will denote the steps of the strategy by(strat1)–(strat5).(strat1) Quasiperiodicity of U mb2 ( x ) (5.11) implies that H ( x ) is also a quasiperiodic oper-ator family, with diagonal entries defined by a new function f :( H ( x )) nn = f ( x + ω · n ) . For sufficiently small ε (depending on the size of R ′ ), the function f will be, say, C reg / x where the original f was C reg -regular, see Corollary 2.4and Remark 2.5.(strat2) As a consequence, our attention should be towards the diagonal entries of H corresponding to singular points for H ( x ). Assume that dist(( S ∪ ( S + e )) , Z d \ R ) ≥
1, so that we avoid the boundary effects. Then, for any m ∈ ( S ∪ ( S + e )), f ( x + ω · m ) is the exact eigenvalue of H ′ R ′ ( x ) that corresponds to the lattice point m . Here, we use the relation between eigenvalues of H ′ R ′ ( x ) and lattice points of R ′ provided in Theorem 4.1.(strat3) Suppose that, for a single block, we have a subset B ⊂ R \ ( S ∪ ( S + e )) thatsatisfies DUCP for S ∪ ( S + e ). For example, if R is a box whose boundary is nottoo close to S ∪ ( S + e ), then we can surround S ∪ ( S + e ) by a layer of thickness twoinside R , see Remark 3.6. For any eigenvalue of H ′ R ′ ( x ), corresponding to a latticepoint m ∈ S ∪ ( S + e ), apply Lemma 3.3 to the corresponding eigenfunction, andconclude that it must be non-trivially supported on B . Since every point of B isregular, the corresponding diagonal entry of the operator is Lipschitz monotone.Proposition 3.10 guarantees that there is a non-trivial positive contribution tothe derivative of each singular eigenvalue from some diagonal entries of B . Thecontribution from the remaining diagonal entries of H ′ R ′ ( x ) to the derivative is alsonon-negative.(strat4) The previous part almost guarantees (conv2), with one caveat: the interpolationprocess in the definition of H ′ R ′ ( x ) creates some x -dependent off-diagonal entries,which also contribute to the derivatives of singular eigenvalues not necessarily in apositive way. However, these entries are outside of R . If we require S ∪ ( S + e ) tobe away from Z d \ R , we can make this contribution arbitrarily small, independentof the lower bound on the contribution from B described above. This will completethe verification of (conv2). The particular selection of the set B does not affectthe definition of U mb2 , it only affects our choice of what to use as a lower boundon the derivatives. Thus, it can be chosen in an x -dependent way, as long as theultimate lower bounds are uniform in x .(strat5) In order to verify (conv3), note that, after diagonalisation, singular points willonly be coupled with points outside of R . The strength of this coupling (i.e, inthe language of Section 2, the length of the corresponding edges) will be in termsof the decay of the corresponding eigenfunctions of H ′ R ′ which, in turn, can beobtained using Proposition 3.7 and Proposition 3.11 (the latter is used to controlthe derivatives, as explained in (conv4)). Ultimately, one can make it bounded by an arbitrarily large power of ε , by requiring dist(( S ∪ ( S + e )) , Z d \ R ) to belarge enough.We will summarize the above construction in the following theorem. Theorem 5.2.
Fix d , C reg , C dio , τ dio , D min ≥ , c sep > . Let also ω ∈ DC C dio ,τ dio satisfy < ω < ω < . . . < ω d < / . Let S ⊂ Z d be a finite subset. There exists r = r ( S ) ∈ N and ε = ε ( d, C reg , C dio , τ dio , c sep , S, E reg , f ) > such that, for < ε < ε and all x ∈ R , the operator H ( x ) satisfies Anderson localiza-tion and is unitarily equivalent to an operator with convergent perturbation series, if thefollowing are satisfied for some x ∈ R : (1) f satisfies ( f and (gen0) . (2) There exists a box R ⊃ S such that the family of covariant moving blocks, con-structed above, satisfies (gen1) – (gen4) . (3) Define R , R ′ as in (5.1) . We will require dist( S ∪ ( S + e ) , Z d \ R ) ≥ r . (4) S sing ( x ) is contained in the support of U mb2 ( x ) , constructed above in (5.10) . Dueto (gen2) , this means that S sing ( x ) is contained in the union of the copies of thesets S ∪ ( S + e ) that appear in the definition of U mb2 ( x ) . (5) The eigenvalues of H S ∪ ( S + e ) ( x ) are uniformly c sep ε -separated for all x ∈ [ x , x + ω ] . Moreover, this property holds with f replaced by f t from (4.1) , uniformly in t ∈ [0 , . Remark 5.3.
There is additional dependence of ε on f , besides the dependence through C reg and E reg . This dependence can be made explicit; we will specify it here in order toavoid overloading the statement of the main result. Consider the smallest box B such thatdist( S ∪ ( S + e ) , Z d \ B ) ≥
4. Then, ε depends on f through the quantity W ( f, M, ω )from Proposition 2.6. Note that the dependence on ω can be reduced to dependence on C dio , τ dio , and the dependence on M is accounted for in the dependence on S . See alsoRemark 3.6. Proof.
We will show that the procedure described in the beginning of the section resultsin an operator unitarily equivalent to H ( x ) and satisfying the assumptions of Proposition2.1. Fix r ≥
4, and take some box R such that dist( S ∪ ( S + e ) , Z d \ R ′ ) ≥ r , where R ′ is defined in (5.1). We will specify additional conditions on r later. Fix some x andconstruct the covariant family of moving blocks, as described earlier in this section. For x ∈ [ x , x + ω ], let B ( x ) be the smallest box in Z d such that(1) B ( x ) ⊃ ( S ∪ ( S + e )), and dist(( S ∪ ( S + e )) , Z d \ B ( x )) ≥ R ⊃ S with dist( Z d \ R, S ∪ ( S + e )) ≥
4, the set { n ∈ B ( x ) : dist( n , Z d \ B ( x )) ≤ } does not contain the point of R ′ with largest absolute value of the potential.The choice of B ( x ) can be made independently of R . Indeed, we can always choosetwo non-overlapping layers of thickness two around S , and then pick one of them withthe smaller maximal value of the potential. For example, let B ( x ) be the smallest box ONVERGENCE OF PERTURBATION THEORY 17 such that dist(( S ∪ ( S + e )) , Z d \ B ( x )) ≥
2, and B ( x ) be the smallest box such thatdist( B ( x ) , Z d \ B ( x )) ≥
2. Then, for any x , we can either pick B = B ( x ) or B = B ( x ).Construct the operator H ( x ) = U mb2 ( x ) − H ( x ) U mb2 ( x )as earlier in the section, using S and R ′ . The diagonal entries of H ( x ) correspond tothe lattice points and are ordered in the same way as the values of the potential. Theentries at distance at least 2 from Z d \ R ′ are exact eigenvalues of the block H R ′ ( x ) (forthe respective lattice points). The corresponding eigenvectors have cluster structure, inthe sense of Proposition 3.7: one cluster would be ( S ∪ ( S + e )) ∩ S sing ( x ), and each of theremaining lattice points of R ′ corresponds to a separate cluster, with separation boundedbelow by a quantity that, ultimately, depends only on r but not on ε . Proposition 3.7implies that each regular eigenvector ψ decays as ψ ( m ) ≤ Cε dist( m , n ) , where n is its support point, and each singular eigenvector decays as ψ ( m ) ≤ Cε dist( m ,S ∪ ( S + e )) . Here, C depends on r , but not on ε . Moreover, the derivatives of the eigenvectors in x satisfy similar decay with a loss of one ε , see Proposition 3.11.As a consequence, each singular diagonal entry of H ( x ) located on S ∪ ( S + e ) hasonly outgoing edges of length bounded by Cε r − , with a derivative bound Cε r − . It wouldbe sufficient to show that the assumption (conv2) is satisfied with µ independent of r .Note that we already have µ = 1 for each regular eigenvalue.In order to establish the above, apply Lemma 3.3 with B = { n ∈ B ( x ) : dist( n , Z d \ B ( x ) ≤ } , A = S ∪ ( S + e ) . Proposition 3.7 implies that any ℓ ( R ′ )-normalized eigenfunction ψ associated to a singularlattice point, satisfies k ψ k ℓ (( S ∪ ( S + e )) ∩ S sing ( x ) ≥ − O ( ε ) ≥ / ε . Lemma 3.3, in view of Remark 3.6 (see also Remark 5.3), impliesthat | ψ ( b ) | ≥ c ε c for some b ∈ B , where c only depends on S and c depends on E reg .Since b is a regular point, Proposition 3.10 implies that the derivative of the singulareigenvalue, corresponding to ψ , has a contribution bounded below by C ε c , where C > S , ω , and E reg . The contributions from all otherdiagonal entries of H ′ ( x ) are also non-negative. Following (strat4), we also need to accountfor the terms associated to the interpolated diagonal entries. These entries are locatedsomewhere in R + ∪ R − , which are outside of R and therefore away from S . By combiningPropositions 3.7 and 3.10, we can see that their contribution to the derivatives of singulareigenvalues is bounded above by a power of ε that grows with r . Thus, as long as, say, r > c + 2, the assumptions are satisfied. (cid:3) One can naturally extend the result of Theorem 5.2 to the case where one requires morethan one type of blocks to cover S sing , assuming that they do not “interact” with eachother. Corollary 5.4.
Let S = { S , . . . , S k } be a collection of subsets of Z d . There exists r = r ( S ) ∈ N such that the following is true.Suppose that the assumptions of Theorem . are satisfied for each S j , with a modificationof (4) that S sing is contained in the union of supports of the corresponding operators U mb ,j ,and those supports do not overlap. Then the conclusion of Theorem . holds. Particular cases and refinements of Theorem 5.2
As the title suggests, in this section we will discuss several more concrete examples of theconstruction from Theorem 5.2. In each particular case, one can make some optimizationsin choosing the unique continuation sets and obtain weaker requirements on the separationbetween different blocks.6.1.
Example 1: the case d = 1 . We will illustrate the above procedure, first, on thecase where f has a single flat segment and is C reg -regular outside of a neighborhood of it.Assume the following.(z1) f ( x ) = E for x ∈ [ a, a + L ] = [ b − L/ , b + L/ ⊂ ( − / , / L is not an integer multiple of ω . Let L = pω + z , p ∈ N , z ∈ (0 , β = min { z, − z } .(z3) f is C reg -regular outside of [ a − β, a + L + β ].(z4) Let M = ⌈ L/ ω ⌉ + 2. Assume that the points b − M ω, b − ( M − ω, . . . , b, b + ω, . . . , b + ( M + 1) ω. are contained inside the interval ( − / , / z
4) is the most important one and has to be present in some form. It manifeststhe separation condition (2) from Theorem 5.2. Let also E reg := max {| f ( b − M ω ) | , | f ( b + ( M + 1) ω ) |} . Theorem 6.1.
Fix C reg , C dio , τ dio , D min = 1 , β > , M ∈ N . Let also ω ∈ DC C dio ,τ dio ∩ (0 , / . Let f satisfy ( f and ( z – ( z with the M , β fixed previously. There exists ε = ε ( C reg , C dio , τ dio , β, M, E reg ) > such that, for < ε < ε , the operator (1.1) is unitarily equivalent to an operator withconvergent perturbation series. As a consequence, H ( x ) satisfies Anderson localization forall x ∈ R .Proof. Let H M ( x ) be the restriction of H ( x ) into the interval [ − M, . . . , M + 1], a (2 M +2) × (2 M + 2)-matrix. The asymmetry is caused by the fact that we will consider H ( x )for x ∈ [ b, b + ω ].For x ∈ [ b, b + ω ], consider the matrix H M ( x ) and modify it as follows: the entries thatcouple − M with − M + 1 will be linearly interpolated from ε at x = b to 0 at x = b + ω ,and the entries coupling M and M + 1, respectively, in the opposite way: 0 at x = b and ε at x = 0. Denote the resulting operator H ′ M ( x ). Denote also by U M ( x ) the diagonalisingoperator obtained in Theorem 4.1 for H ′ M ( x ). Proposition 4.4 combined with Proposition3.9 guarantees that the family U M ( x ) satisfies the assumptions of Theorem 4.1 (note thatwe cannot apply 4.4 directly, since, after rescaling, the range of energies will be of the size ONVERGENCE OF PERTURBATION THEORY 19 ε − ; however, one can use 3.9 to treat large eigenvalues corresponding to near the edgesof the interval).Denote by U mb0 ( x ) the matrix U M ( x ) extended as identity into ℓ ( Z ). At the moment,it has only been defined for x ∈ [ b, b + ω ]. However, the block structure of U mb0 ( x ) implies(for a single value of x = b for which the equality makes sense):(6.1) U mb0 ( x + ω ) = T − U mb0 ( x ) T. As in the previous section, we will use (6.1) as the definition of U mb0 ( x ) for x ∈ R . Wenow perform the above procedure with each “copy” of the singular interval appearing in H ( x ). That is, define U mb1 ( x ) = Y m ∈ Z U mb0 ( x + m ) . Note that (z4) guarantees that the supports of the factors in the above product aredisjoint. We do not need to introduce U mb2 , since there are no other lattice directions.Now, let H ( x ) = U mb1 ( x ) − H ( x ) U mb1 ( x ) . At x = a , the diagonal matrix elements of H ( a ) at [ − M + 3 , M −
3] correspond to thevalues of f calculated on the flat segment. They are exactly equal to the eigenvalues of H ′ ( a ). In the notation of the previous section, we have S = [ − M + 3 , M − x goes from a to a + ω , the position of the flat segment on Z shifts by 1: the entrylocated at − M + 3 becomes regular, and the one located at M − − M + 2 , M − − M, − M +1 , − M + 2 , M − , M, M + 1 are always regular and are small perturbations of the entriesof H ( x ), and therefore inherit their monotonicity properties. The remaining eigenvaluescorrespond to the set [ − M + 3 , M + 2], which will play the role of S ∪ ( S + e )) from theprevious section: S ∪ ( S + e ) = [ − M + 3 , M − . For each x ∈ [ b, b + ω ], consider the points(6.2) x − M ω, x − ( M − ω, . . . , x, x + ω, . . . , x + ( M + 1) ω. Assumption (z2) guarantees that both endpoints of [ a, a + L ] cannot be close to the points(6.2) at the same time. More precisely, either a or a + L is at least β/ p, p + 1 ∈ [ − M, M + 1]such that x + pω and x + ( p + 1) ω are two closest points to [ a, a + L ] on one side, but stillnot too close (both are β/ x = b , then the points p = − M + 1, p + 1 = − M + 2 would work. As x increases, we may have to switch to a pair of pointson the other side. The set { p, p + 1 } will play the role of B , the unique continuation setfrom the previous section. As discussed previously, this set is x -dependent; however, allcases can be treated similarly, and we will assume B = {− M + 1 , − M + 2 } for simplicity.Let ψ be an eigenfunction corresponding to one of the lattice points on S ∪ ( S + e ) = [ − M + 3 , M − ℓ ([ − M, M + 1]), with eigenvalue E j ( x ), j ∈ [ − M + 3 , M −
2] (here, the eigenvalues are identified with lattice points throughTheorem 4.1). By Proposition 3.7, ψ is supported on [ − M + 3 , M −
2] up to O ( ε ).Moreover, ψ ( − M + 2) = O ( ε ), ψ ( M + 1) = O ( ε ). The values f ( x + mω ), m ∈ S , are allequal to E , except maybe for f ( x + ( M − ω ). We will now discuss the unique continuation procedure from B = {− M + 1 , − M + 2 } into [ − M + 3 , M − E j +1 ( x ) − E j ( x ) ≥ c ( ω, M ) ε. On the other hand, by considering the eigenvalues as perturbations of the diagonal ele-ments, we also have | E j ( x ) − E | ≤ ε, j ∈ [ − M + 3 , M − | E M − ( x ) − f ( x + ( M − ω ) | ≤ ε. In both cases, the right hand side is actually 2 ε + c ( ω, M ) ε , and we assumed ε to besmall enough for simplicity. Assume first that ψ is a singular eigenfunction correspondingto E j with j ∈ [ − M + 3 , M − | ψ ( − M + 2) | ≥ c ( M, ω, E reg ) ε. Indeed, if | ψ ( − M + 2) | is too small, then since | ψ ( − M + 1) | = O ( ε ), iterating theeigenvalue equation will contradict the ℓ -normalization condition). Apply Proposition3.10 and conclude: E ′ j ( x ) ≥ c ( M, ω, E reg ) ε , j ∈ [ − M + 3 , M − . For the remaining eigenvalue E M − , we need to consider two cases. If, say, | E − f ( x +( M − ω ) | < ε , then the previous argument also applies to the eigenfunction correspondingto E M − . If | E − f ( x + ( M − ω ) | ≥ ε , then the point M − | ψ ( M − | ≥ k ψ k ℓ ). Then, one can apply Proposition3.10 directly at that point and obtain an even stronger conclusion: E ′ M − ( x ) ≥ c ( M, ω, E reg ) > . In other words, such point can be considered as a regular point for practical purposes.The above argument implies (conv2) from Proposition 2.1 with µ = 2 for our operator.We will illustrate (conv3) on the example with M = 5. The result of applying Proposition3.7 to the eigenvectors of H ′ M ( x ) can be described as follows: U M ( x ) ≈ ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε εε ε ε ε ε ε ε ε ε ε ε An entry ε k means that the corresponding component of the eigenvector is bounded aboveby cε k , where c does not depend on ε . The above matrix illustrates the worst possiblecase for x ∈ [ a, a + ω ]: for example, at least one entry in edge of the central block is, ONVERGENCE OF PERTURBATION THEORY 21 actually, ε instead of 1, and some of the remaining entries in that column are actuallybetter by one order of ε . Remark 3.11 implies that the derivatives in x of each eigenvectorare at most ε worse than the actual components. The central 6 × − , U M ( x ) diagonalises a block of H ′ ( x ) rather than H ( x ), the conjugation of H ( x )by U mb0 ( x ) will result in some extra off-diagonal entries. To calculate them, let us subtract H ′ ( x ) from H ( x ) and calculate the result of the conjugation of the remaining part: U M ( x ) T
00 0 1 · O (1) ε ε ε ε × × × × × ε
00 0 0 0 ε ε ε O (1) · U M ( x ) 00 0 1 . Each element of the matrix product is made out of products of three entries, one from eachmatrix. Even without multiplying three (14 × − ,
3] block, oneneeds to use a matrix entry from the right factor that connects point entry from [ − , − , ε . If we gained exactly ε , then one will gain at least one ε from the central factor. Thus, any jump that startsinside of the [ − ,
3] block will cost at least ε , which is sufficient to satisfy (conv3) andapply Proposition 2.1 (in view of (conv4) and Remark 3.11). (cid:3) Remark 6.2.
One can check that, under the assumptions of Theorem 6.1, the derivativeof the integrated density of states of the family H ( x ) has a spike of height ≍ ε − andwidth ≍ ε around energy E . One can produce larger spikes by combining multiple flatintervals at different energies, see Examples 5 and 6 below.6.2. A higher-dimensional version.
The approach from the previous subsection canbe extended to several higher-dimensional examples.
Example 2:
Let d ≥
2. Suppose that f and ω satisfy all assumptions from the previoussubsection (with ω = ω ). Additionally, assume the following(z5) For any x ∈ [ a, a + L ] and any n ∈ Z d , n / ∈ Z e , | n | ≤
6, we have f being C reg -regular at x + n · ω . Theorem 6.3.
Suppose that the assumptions of Theorem . are satisfied with ω replacedby ω . Assume also that ω ∈ DC C dio ,τ dio and ( z . Then there exists ε = ε ( d, C reg , C dio , τ dio , β, M, E reg ) > such that the operator (1.1) satisfies Anderson localization on Z d .Proof. We only outline the changes in the proof. Instead of the interval [ − M, M + 1],consider the box [ − , d − × [ − M, M + 1]and construct U M ( x ) by applying Theorem 4.1 to the above box. One will need tointerpolate the Laplacian terms that couple [ − , d − × {− M } and [ − , d − × {− M + 1 } between ε and 0, and the terms coupling [ − , d − × { M } and [ − , d − × { M + 1 } between 0 and ε , as x goes from b to b + ω . The eigenvalues on the “central” box { } d − × [ − M, M + 1] are cε -separated due to Proposition 4.4. Eigenvalues correspondingto the remaining entries are separated from each other by a constant independent of ε for ε ≪ e will shift the eigenvalues of the block { } d − × [ − M, M + 1] by atmost O ( ε ) due to Proposition 3.9, and therefore will preserve O ( ε )-separation property.The rest follows from the proof of Theorem 6.1. One needs to specify the uniquecontinuation subset. For x = a it will be B = (cid:0) { } d − × {− M + 1 , − M + 2 } (cid:1) ∪ (cid:0) { n ′ ∈ Z d − : | n ′ | = 1 } × [ − M + 1 , M − (cid:1) . In other words, we are allowed to make up to two steps to the left from the singular subset,and one step in one other direction. The set B can be imagined as a “claw” surroundingthe singular set S .Proposition 3.7 implies that any singular eigenfunction restricted to {− M + 1 } × { n ′ ∈ Z d − : | n ′ | = 1 } will be O ( ε ). The rest of the set B is one step away from S , and thereforeany eigenfunction will have, ultimately, at least cε of mass somewhere on B . One mayhave to treat the right-most point of S separately, the argument does not change fromTheorem 6.1, in view of Proposition 4.4 and Proposition 3.9.Once U M ( x ) is constructed for x ∈ [ b, b + ω ], we repeat the steps in Section 5 in order toconstruct the operator U mb2 . Condition (z3) will guarantee that the supports of differentcopies of U mb0 ( x ) will not overlap. (cid:3) Several additional cases.
In this subsection, we outline several additional caseswhich, ultimately, are covered by Theorem 5.2.
Example 3:
One can consider multiple intervals of the type described in Theorem 6.1,as long as they are separated enough so that the supports of the operator U are disjoint.Similarly, one can combine multiple “non-interacting” cases of Theorem 6.3. Example 4:
More interestingly, if two intervals from Theorem 6.1 are not sufficientlydisjoint, one can consider them as one singular set and apply the argument from thegeneral theorem. One can also do it in the setting of Theorem 6.3, with a strengthenedversion of (z5).
Example 5:
The following example can be called “a chain of intervals”. Let d = 1 andLet I , . . . , I N ⊂ ( − / , /
2) be a collection of disjoint intervals: f ( x ) = E j for x ∈ I j ; E j < E k for j < k, and for j = 1 , . . . , N −
1. Assume also that, for sufficiently large r = r ( N ) ( r = 2 N shouldbe enough), we have − / < I − rω < I N + rω < / . Then the assumptions of Theorem 5.2 are satisfied, and we have Anderson localizationfor small ε . Remark 6.4.
Suppose that, on top of all these assumptions, that I j +1 = I j + ω ; and wealso assume for simplicity that N = 2 k −
1. Then it is easy to see that f ′ ( x ) ≍ ε k for x ∈ I k . As a result, the derivative of the integrated density of states has a spike of height ≍ ε − k and width ≍ ε k around E k . ONVERGENCE OF PERTURBATION THEORY 23
Example 6:
The following example is a combination of Examples 2 and 5. It can becalled “a tree of intervals”. Let I , . . . , I N be a collection of disjoint intervals: f ( x ) = E j for x ∈ I j ; E j = E k for j = k. Suppose that for some x ∈ I we have, say, x := x + ω ∈ I . Additional shifts by ω , . . . , ω d may produce points, say, x = x + ω ∈ I , . . . . Assume that, for any x ∈ I ,we are guaranteed to escape all intervals after, say, q steps in the directions perpendicularto e , and would only be able to come back to I after additional r steps. In other words,for x ∈ I , f is regular at x + ω ′ · n ′ , for all n ′ ∈ Z d − , q ≤ | n ′ | ≤ q + r . One canrefine this in each particular case, if the number of steps needed to escape the system ofintervals depends on the direction. In all cases, we would require a “layer” of thickness r that consists of regular points and surrounds the collection of singular points describedabove. The thickness of the layer will depend on the number of steps required to escape thesingular intervals. For r sufficiently large, the assumptions of Theorem 5.2 will be satisfied.In particular, to establish the separation condition (5) from Theorem 5.2, note that eachinterval produces a block with cε -separated eigenvalues due to Proposition 4.4. Differentintervals are located at different energies, and therefore do not affect each other (eachlies in its own cluster). The coupling between these intervals satisfies the assumptionsof Proposition 3.9, and therefore also does not affect the separation condition. After theconjugation, we will have, for the new diagonal function f ( x ), f ′ ( x ) ≍ ε µ j , x ∈ I j , where µ j depends on the number of steps in the directions perpendicular to e requiredto escape the interval (by more careful analysis, one can show that it is equal to twice thenumber of steps). As a consequence, the derivative of the integrated density of states of H will have spikes of order ε − µ j around the energies close to E j . References [1] B´ellissard J., Lima R., Scoppola E.,
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Email address : [email protected] Department of Mathematics and Computer Science, St. Petersburg State University,14th Line 29B, Vasilyevsky Island, St. Petersburg 199178, Russia
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