THE SPECTRAL ANALYSIS OF SCHRÖDINGER OPERATOR ON GENERAL GRAPHS. Lukun Zheng

Transcription

1 THE SPECTRAL ANALYSIS OF SCHRÖDINGER OPERATOR ON GENERAL GRAPHS by Lukun Zheng A dissertation submitted to the faculty of The University of North Carolina at Charlotte in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics Charlotte 205 Approved by: Dr. Stanislav Molchanov Dr. Boris Vainberg Dr. Yuri Godin Dr. Susan Sell

2 c 205 Lukun Zheng ALL RIGHTS RESERVED ii

3 iii ABSTRACT LUKUN ZHENG. The spectral analysis of Schrödinger operator on general graphs. (Under the direction of DR. STANISLAV MOLCHANOV) The goal of this dissertation is to give the sufficient conditions for the absence of a.c.spectrum or existence of the pure point (p.p.) spectrum for the deterministic or random Schrödinger operators on the general graphs. For the particular situations of non-percolating graphs like Sierpiński lattice and Quasi- dimensional tree, we ll prove the Simon-Spencer type results and the localization theorem for Anderson Hamiltonians. Technical tools here are the extensions of the real-analytic methods presented for the D lattice Z and corresponding Schrödinger operators in [0]. The central moment is the cluster expansion of the resolvent with respect to appropriate partition of Γ.

4 iv ACKNOWLEDGMENTS Upon the completion of this thesis I would sincerely gratefully express my thanks to many people. First of all I would like to show my greatest respect and gratitude to my supervisor, Dr. Stanislav Molchanov who was greatly helpful and offered invaluable guidance to me on both academic performance and personal life during my study at University of North Carolina at Charlotte. The experience working with him will go forever with me in my life. His attitude to research work and his attitude to life deeply engraved in my heart and memory. Special thanks also go to the members of the supervisory committee, Dr. Boris Vainberg, Dr. Yuri Godin and Dr. Susan Sell without whose solid wisdom and abundant assistance I would not have accomplished my doctoral study. Also I would never forget the generous support from Dr. Zhiyi Zhang and Dr. Jiancheng Jiang. Not forgetting to my honourable professors in the Department of Mathematics and Statistics who supported me on such an unforgettable and unique study experience for five years. Here allow me to express my full love and gratitude to my beloved families. Speechless thanks for their understanding, their support and their love during so many years from my birth. Also, I would like to thank my friend Xing Zhang who is always there whenever I need any kind of help. He give me all that can be expected from a friend: support, trust, loyalty, respect, company, and so on. Finally, deep gratitude also due to my graduate friends through the duration of my study in this university. Thank all of them for sharing their time with me and priceless assistance.

5 v TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES vi vii CHAPTER : INTRODUCTION.. A General Summary.2. Two Particular Examples 3 CHAPTER 2: BASIC DEFINITIONS AND THE MODEL 5 CHAPTER 3: Expansion Theory of the Resolvent Kernel Path Expansion Cluster Expansion 9 CHAPTER 4: Technical Lemmas based on the resolvent kernel R (x 0, x) where Im() = ɛ > 0 CHAPTER 5: The real analytic approach to the absence of a.c. spectrum and localization 5 2 CHAPTER 6: Theorems on the absence of a.c. spectrum and localization 25 CHAPTER 7: Examples 33 REFERENCES 37

6 vi LIST OF FIGURES FIGURE : Sierpiński lattice S 4 FIGURE 2: Quasi- dimensional tree T 4 FIGURE 3: Main blocks and belts FIGURE 4: The path [γ]

7 LIST OF TABLES vii

8 CHAPTER : INTRODUCTION. A General Summary The idea on the possible connection between the spectral phase transition for the Anderson Hamiltonian on the lattices Z d, d 3 and the percolation was always popular in the physical literature. In one direction it is definitely correct. Consider on the lattice Z the random Schrödinger operator Hφ(x) = φ(x) + σv (s, ω), φ(x) = φ(x + ) + φ(x ) where σ is a coupling constant and V (x, ω) are unbounded i.i.d., say, N(0, ) random variables. Then for arbitrary small σ and for arbitrary large M there are infinitely many points x : V (x, ω) > M. I.e. the set {x : V (x, ω) M} contains only finite connected components, or they doesn t percolate, though its concentration can be arbitrary close to. We call Z non-percolating graph. More formally, let Γ be an abstract graph and X(x, ω), x Γ is the Bernoulli field such that P {X( ) = } = ɛ, P {X( ) = 0} = ɛ. We call Γ a non-percolating graph if for arbitrary ɛ > 0 the set {x : X(x) = 0} contains P a.s. only bounded connected components. The class of non-percolating graphs is very rich. It includes, for instance, all nested fractal lattices, among them, the Sierpiński lattice.

9 2 Let s consider for such graphs Anderson type operators Hφ(x) = φ(x ) + σv (x, ω) x :x x where {x : x x} is the set of nearest neighbours of x and V (x, ω) are unbounded i.i.d. random variables. At the level of the physical intuition we have the following picture: the realization of σv ( ) contains the sequence of the higher and higher walls and the quantum particle can t avoid the interaction with such walls in its attempts to reach infinite. Since the tunnelling through higher and higher walls has smaller and smaller probability one can expect here some kind of localization phenomena, say, the absence of the a.c. spectrum. In the case of D lattice Z these a bit fuzzy arguments were transformed in the mathematical theorem in the famous Simon-Spencer theorem [3]. Let Hφ(x) = φ(x + ) + φ(x ) + V (x)φ(x), x Z. If the potential V (x) is unbounded near ± (i.e. lim sup x + + ), then Σ a.c. =. V (x) = lim sup V (x) = x Of course, for the random i.i.d. unbounded potentials this results provides the absence of a.c. spectrum P a.s.. The Simon-Spencer theorem can be extended on some class of the potentials even in R d, d 2, see [8]. All results of such kind include the very strong assumptions of the existence of the infinite system of the rings or belts b n around origin, where potential V is higher and higher (i.e. min x b n V (x) = h n +, n ). They also require the additional conditions on fast increasing of the heights h n. B.Simon [3] constructed (for Γ = Z 2 ) such Schrödinger operator, where

10 3 V (x) 0 contains the system of the higher and higher walls (i.e. set {x : V (x) M} doesn t percolate for any M), but spectrum of H contains the a.c. components. Unfortunately the lattice Z d, d > percolate, there exist the critical thresholds h cr, h cr : if h > h cr then the set {x : V (x, ω) > h} doesn t percolate, but for h < h cr it contains the infinite component. The goal of this paper is to give the sufficient conditions for the absence of a.c.spectrum or existence of the pure point (p.p.) spectrum for the deterministic or random Schrödinger operators on the general graphs. For the particular situations of nonpercolating graphs we ll prove the Simon-Spencer type results and the localization theorem for Anderson Hamiltonians. Technical tools here are the extensions of the real-analytic methods presented for the D lattice Z and corresponding Schrödinger operators in [0]. The central moment is the cluster expansion of the resolvent with respect to appropriate partition of Γ..2 Two Particular Examples The general theory provide the following particular results. Theorem (A). Consider the Sierpiński lattice S (see figure ) with Laplacian ψ(x) = ψ(x ) and boundary condition ψ(0) = 0. Let H = + V (x, ω) is the Anderson x x Hamiltonian and V (x, ω) are i.i.d. unbounded random variables. Then P a.s. H = H(ω) has no absolutely continuous spectrum. 6. This theorem is a particular case of the much more general result, see Corollary Theorem (B). Consider the Quasi- Dimensional Tree T (see figure 2) with Laplacian

11 4 Figure : Sierpiński lattice S. Figure 2: Quasi- dimensional tree T ψ(x) = x x ψ(x ), x (, 0) and boundary condition ψ(, 0) = 0. Let H = + V (x, ω) be the Anderson Hamiltonian and V (x, ω) are i.i.d. random variables with bounded distribution density f(t)( f(t) < M, t R, for some M < ) and f(t)dt > 0 for any A > 0. Then the spectrum of H is pure point with probability t >A.

12 CHAPTER 2: BASIC DEFINITIONS AND THE MODEL Let (Γ, A) be a undirected infinite graph with symmetric adjacency matrix A(Γ) = A (Γ) = [a(x, y)]. If a(x, y) =, we will call x and y are the nearest neighbors, denoted by x y. Suppose that for each x Γ, the number ν(x) of nearest neighbors x : x x is bounded: ν(x) K for some fixed K : 2 K. For the homogeneous graph (with appropriate group of the symmetries) ν(x) K. In this case, K is called the index of branching of the graph (Γ, A). Typical examples of homogeneous graphs are: lattices Z d with index of branching equal to 2d, groups with finite number of generators, homogeneous trees, etc. The path [γ] of length [γ] = n, from point x to point y on Γ, denoted by [γ] : x y is defined to be a sequence of the points [γ] = {x i } n i=0 such that x i x i for i =, 2,, n and x 0 = x, x n = y. Here x 0 = x is called the start point, x n = y is called the end point. All the other points are called the internal points of the path. We denote (γ) : x y the internal part of [γ] : x y, that is, (γ) = {x i } n i= with x x 0 = x and x n x n = y. If a path contains infinitely many points, then it is called an infinite path. Any non-empty set B V with more than one points

13 is called connected or -connected if x, y B, [γ] : x y and [γ] B. The boundary of B is defined as: 6 B = {y : y / B, and y x, for some x B}. In this paper, we assume that graph Γ is -connected. A metric d(x, y) on Γ and distances d(x, B), d(b, B 2 ) are defined in the standard way. The volume of the ball, centered at point x 0 Γ, can not increases faster than an exponential: B R (x 0 ) = {x : d(x, x 0 ) R} K R +. Recall that K = max x Γ ν(x). Let l 2 (Γ) be the Hilbert space of square-summable functions f(x) : Γ C with the inner product and norm (f, g) = x Γ f(x)g(x), f 2 = x Γ f(x) 2. () for f, g l 2 (Γ). The lattice Laplacian in the space l 2 (Γ) is given by a usual formula As easy to see, f(x) = = i.e. the lattice Laplacian is bounded. x :d(x,x)= sup f K. f: f = f(x ) (2)

14 7 The Schrödinger operator(hamiltonian), by the definition, has a form H = + V (x) (3) where V (x) is an arbitrary real-valued potential. In the most interesting case, V (x) = σξ(x, ω) and ξ(x, ω) will be the family of i.i.d.r.v. with bounded continuous density p ξ ( ). Here x Γ, ω (Ω, F, P )(a basic probability space), and σ > 0 is the coupling constant(the measure of disorder). In this case, we will call H the Anderson Hamiltonian on l 2 (Γ). The fundamental and still unsolved problem is to determine the spectral type of H for the general graphs ( or at least for lattices Z d, d 2). It is known ([6])that for arbitrary graph Γ and the very general symmetric bounded operators L, the spectral measure is pure point(p.p) for the large disorder, σ > σ 0, where σ 0 can be effectively determined by the geometry of the graph Γ. If Γ = Z ( or Γ = Z A, Card(A) < ) the spectrum of H is p.p. P-a.s. and the corresponding eigenfunctions are exponentially decaying for arbitrary small coupling parameter σ ( see details in [9]). The second case where the spectral picture is well understood is the homogeneous tree T N, see [5]. The last case is the only one which demonstrate the Anderson type phase transition from the p.p. spectrum for large σ to the mixed spectrum (a.c. component plus p.p. component) for small σ.

15 CHAPTER 3: EXPANSION THEORY OF THE RESOLVENT KERNEL 3. Path Expansion The resolvent kernel of the operator H R (x, y) = (H I) (x, y) is well defined at least for complex. We will use the following exact formula for R (x, y)(so-called path expansion) see [0]. Proposition. Let V (x) be the potential of the Schrödinger operator on Γ with Range(V ) = {V (x) : x Γ} R. Then R (x, y) = δ y(x) V (x) + [γ]:x y ( z γ ). (4) V (z) where δ y (x) = if x = y and 0 otherwise. This formula holds, at least for s such that d(, Range(V )) K + δ, for some δ > 0. Proof. Since Hψ(x) ψ(x) = δ y (x), we have Thus, x :x x ψ(x) = ψ(x ) + V (x)ψ(x) ψ(x) = δ y (x). δ y(x) V (x) + x :x x ψ(x ) V (x). For each x, one can use the same formula, and continue these iterations, to get

16 9 (4). The number of paths [γ] with the fixed start point x and the length n is at most K n and V (z). These facts lead to the convergence of the (K + δ) [γ] + series (4). z [γ] Similar construction works for the restriction of H on subsets B of Γ. Consider H B = + V (x) with Dirichlet boundary condition: φ(x) = 0 for x Γ B. Proposition 2. Let R (B) (x, y) = (H B I) (x, y) be the resolvent kernel of H B (which is well defined at least for Im K + δ, δ > 0 ). Then for x, y B, R (B) (x, y) = δ y(x) V (x) +. (5) V (z) [γ]:x y [γ] B Note that for cardinality of the set B: B < the spectrum of H B contains B real eigenvalues (which can be multiple), i.e. R (B) (x, y) is real and finite for all real, except the finite set of eigenvalues of H B. z [γ] 3.2 Cluster Expansion Let us fix some point x 0 Γ, called reference point. A set b Γ is called a belt with respect to x 0 if any path that starts from point x 0 and goes to infinite intersects with set b. Define the set of all the points trapped by the belt b the enclosure E b of the belt b: E b = {x : [γ] : x 0 x, and [γ] b = }. Thus E b is -connected, that is, x, y E C, [γ] : x y, [γ] E b. The inner boundary of the belt b is defined to

17 0 be the set b = b E b and the outer boundary of the belt b is b + = b b. A counter is defined to be a belt c for which every point in c is irreducible. In other word, c is not going to be a belt if any point of it is taken away. We can define now the following quantity (similar to resolvent of b): for x b, y b +, put β b (x, y) = [γ]:x y (γ) b z (γ) V (z). (6) (and similar expression if x b +, y b ). Note that path (γ) stays inside belt b. In the future the potential V ( ) will be large on b, i.e. for I (I is a fixed interval) the quantity β b (x, y) will be small. Physically it means that the tunnelling of the quantum particle through the belt is very unlikely. Now assume that {b i : i } is a sequence of disjoint belts with respect to the reference point x 0 and the corresponding sequence of the enclosures is {E bi } i=. Assume that E bi E bi+, for any i =, 2, 3,. Let S 0, S, be subsets of Γ between successive belts(s 0 contains x 0 ). That is, S i = E bi E bi b i for i =, 2,. We define the main blocks (0), (), as: (0) = S 0 b, (0) = b +, () = b S b 2, () = b b + 2 and so on. See figure (3) We call b i the inner boundary of b i and b + i the outer boundary of b i for i. Now we will study the cluster expansion of R (x 0, x). Consider the path expansion of H: R (x 0, x) = δ x 0 (x) V (x 0 ) + [γ]:x 0 x z [γ] V (z)

18 Figure 3: Main blocks and belts Figure 4: The path [γ] and introduce the procedure of the union of the paths into the special groups(clusters). Consider a particle following some path [γ] : x0 x (Figure 4). It will stay in the main block (0) for certain amount of time. And then at some moment it will reach b +. By definition, it is the moment of transition from main block (0) to main block (). The particles again can stay for certain amount of time in (0) () doing transitions between (0) and () until at some moment it reaches b + 2, and this moment, by definition, is the transition from () to (2), etc. Therefore, for any [γ] : x0 x, one can define the sequence of the main blocks with which [γ] successively intersects. Let s introduce the graph Γ with the vertices

19 being the main blocks: (0), (),... and possible transitions (n) (n ± ), n > 0 and (0). For each path [γ] on Γ, one can consider the cluster of the elementary paths [γ] following [γ] between main blocks. If so, we say that [γ] [γ]. The class of all the paths which stays in (0) forever corresponds to the [γ] = (0). Put [γ] R (x 0, x) = ( [γ] [γ] z [γ] V (z) Here the [γ]s under summation are from x 0 to x in the kth main block (k). By definition, we have R (x 0, x) = δ x 0 (x) V (x 0 ) + [γ]:(0) (k) ). [γ] R (x 0, x). Let s provide a deeper analysis of any path [γ] [γ]. Assume that [γ] contains, say, 2 transitions (0) () (2). Then at the moment τ + the path enters to some point z 2 on b + for the first time ( the first entrance to () ). Let τ is the last moment before τ + when the path enters to the belt b through b and stays in b until the moment τ +. After moment τ +, the trajectory [γ] is moving inside () and at the moment τ + 2 enters to b + 2 ( the first entrance to (2) ). τ 2 is similarly defined as τ so that in the time interval [τ 2, τ + 2 ] it stays inside b 2, etc. This classification leads to the simple combinatorial representation of R [γ] (x 0, x). For instance, consider the special case when x = x 0. If [γ] [γ] (0), R (x 0, x 0 ) = R (0) (x 0, x 0 ). For other paths [γ], say, contribution to R (x 0, x 0 ) equals [γ] = (0) () (2) () (0), their

20 3 [γ] R (x 0, x 0 ) = R (0) (x 0, z )β b (z, z 2 )R () (z 2, ) R (2) (, z 2l 3)β b 2 (z 2l 3, z 2l 2 )R () (z 2l 2, z 2l )β b (z 2l, z 2l )R (0) (z 2l, x 0 ) ( ) where the summation is over z b, z 2 b +,..., z 2l 3 b + 2, z 2l 2 b 2, z 2l b +, z 2l b with l being the number of times the path go through the belts until it returns back to x 0. For each term R (k) (z i, z j ), k =, 2,, there are four possible versions according to the different transitions through the kth belt, i.e. R (k) (z i, z j ), z i b + k, z j b k+ ; R(k) (z i, z j ), z i b + k, z j b + k ; R (k) (z i, z j ), z i b k+, z j b k+ ; R(k) (z i, z j ), z i b k+, z j b + k. For each term R (0) (z i, z j ), there are two possible versions: R (0) (x 0, z j ), z j b ; R (0) (z j, x 0 ), z j b. The similar formulas have place for R [ γ] (x 0, x), x (k). Say, [ γ] = (0) () (k + ) (k), [γ] R (x 0, x) = R (0) (x 0, z )β b (z, z 2 )R () (z 2, z 3 )β b 2 (z 3, z 4 )R (2) (z 4, ) R (k+) (, z 2l )β b k+ (z 2l, z 2l )R (k) (z 2l, x) where the summation is over all z b, z 2 b +, z 3 b 2, z 4 b + 2,..., z 2l b + k+, z 2l b k+ with l being the number of times the path go through the belts until it reaches to x.

21 For convenience of the reference, let s formulate the main result of this section as 4 Theorem 3 (The cluster expansion theorem). Let x 0 be the reference point and x (k), then Here R (x 0, x) = δ x 0 (x) V (x 0 ) + [γ]:(0) (k) [γ] R (x 0, x). (7) [γ] R (x 0, x) = R (0) (x 0, z )β b (z, z 2 )R () (z 2, z 3 )β b 2 (z 3, z 4 )R (2) (z 4, ) (, z 2l )β b k+ (z 2l, z 2l )R (k) (z 2l, x) (8) R (k+) where the summations are described above.

22 CHAPTER 4: TECHNICAL LEMMAS BASED ON THE RESOLVENT KERNEL R (X 0, X) WHERE IM() = ɛ > 0 In this chapter we will formulate and prove several criteria for the absence of the a.c. spectrum, localization, etc. All results here will be based on the complex analysis. See real analytic approach in Chapter 5. Due to general theory, H = E(d). If f l 2 (Γ), then µ f (d) = (E(d)f, f) SpH is the spectral measure of the element f and µ f (d) = µ f (d) = f 2 2. R Sp(H) The resolvent R = (H I) is the bounded operator for / Sp(H), for instance, for = 0 + iɛ, 0 R µ f (dz), (R f, f) = z. The function f 0(x) l 2 (Γ) has Sp(H) the maximal spectral type if µ f0 (d) µ g (d) for any g l 2 (Γ). The dense set of functions f 0 l 2 (Γ) have the maximal spectral type. The measure µ f (d) has the Lebesgue decomposition into a.c. component µ f,a.c., singular continuous component µ f,s.c., and finally point component µ f,p. The a.c. components is responsible for the transport of the quantum particles (electric conductivity, scattering, etc). The point component is related to the localization phenomena. Theorem 4. The limit π lim ɛ 0 Im(R +ɛi f, f) exists for a.e. R and equals to ρ f (). Here ρ f () is the density of the a.c. part of the spectral measure µ f (d). See details in [2].

23 Corollary 5. For given energy interval I R, the a.c. part of the spectral measure µ f,a.c. (d) is equal to 0 if a.e. on I. lim Im(R +ɛi f, f) = 0 ɛ 0 6 Assume that µ a.c. (d) = 0 (operator H has no a.c. spectrum). How to separate µ p and µ s.c.? The following theorem (see [4]) gives the simple criteria for (p.p) spectrum. Theorem 6 (Simon-Wolff). Assume that for real on the resolvent kernel, R +ɛi (x 0, y) ɛ 0 R +0i (x 0, y) and for any x 0 Γ lim ɛ 0 y Γ R +iɛ (x 0, y) 2 = y Γ Then the operator H typically has p.p. spectrum. R +0i (x 0, y) 2 <. Remark. It is not difficult to prove the existence of the limit above. The last sentence typically means that, in the subspace l 2 (δ x0 ) = Span(R δ x0 ), the perturbed operator H a = H + aδ x0 ( ), x 0 Γ (rank-one perturbation) has p.p. spectrum for a.e. a R. At the same time(in the wide class of situations), the operator H a has the pure singular spectrum for some a from the appropriate G δ set (with measure 0), see []. This result is especially convenient for the random Schrödinger operators (Anderson Hamiltonians): H(ω) = + σξ(x, ω), x Γ, ω (Ω, F, P ), where σ is a coupling constant (the measure of disorder) and the random variables ξ(x, ω) are i.i.d. with bounded continuous distribution density.

24 7 As we already mentioned, for large σ (large disorder) the operator H(ω) has p.p. spectrum P-a.s., see [6]. Let s formulate and prove the result, closely related to Simon-Spencer approach to the theorems on the absence of the a.c. spectrum. Theorem 7. Let Γ be a graph with estimate ν(x) K, is the Laplacian, H = + V (x) is the Schorödinger operator with potential V ( ). Assume that for some sequence of points D = {x n : n =, 2,... } n= V (x n ) <. Consider the new operator H = + V (x) with boundary condition φ(x n ) = 0, n =, 2,..., i.e. the restriction of H on Γ D with Dirichlet boundary condition on D. Then Sp a.c. (H) = Sp a.c. ( H) and for any f L 2 (Γ), the a.c. components of the spectral measures of f for H and H are mutually a.c.. In particular, Sp a.c. ( H) = Sp a.c. (H) =. Proof. The proof is based on Kato-Birman criterion []. See also [2, 3]. If for some fixed 0 = Ci, the difference between the resolvents R 0 = (H 0 I) and R 0 = ( H 0 I) belongs to the trace class, i.e. R 0 (x, x) R 0 (x, x) <, x Γ then µ a.c. (H) = µ a.c. ( H) Now we will use the path expansion to calculate R 0 (x, x) and R 0 (x, x), x Γ for

25 8 0 = K(K + )i. We have for x Γ, and R 0 (x, x) = 0 V (x) + ( [γ]:x x z [γ] 0 V (z) ) R 0 (x, x) = 0 V (x) + ( [γ]:x x z [γ] [γ] D= 0 V (z) and R 0 (x, x) = 0, x D. Note that 0 V (z) K(K + ). If x n D, then ), x Γ D R 0 (x, x) R 0 (x, x) = R 0 (x, x) ( ) K 0 V (x n ) + K(K + ) + K 2 (K(K + )) V (x n ) 2 V (x n ). Therefore, R 0 (x, x) R 0 (x, x) 2 V (x n ) x D n= <. (9) But if x / D, we have R 0 (x, x) R 0 (x, x) = ( [γ]:x x z [γ] [γ] D 0 V (z) ). Since [γ] D =, the path [γ] must intersect D for the first time at one of the points x, x 2,.... Let A j = {[γ] : x x and γ and D intersect for the first time at x j }, j =, 2,....

26 9 Then ( γ A j z [γ] 0 V (z)) γ A j (K(K + )) γ V (x j ) ( V (x j ) ) K d(x,x j) (K(K + )) + K d(x,xj)+ d(x,x j) (K(K + )) +... d(x,x j)+ ( + ). K V (x j )(K + ) d(x,x j) Therefore, for x / D, But R 0 (x, x) R 0 (x, x) x Γ D R 0 (x, x) R 0 (x, x) x Γ D ( + ) K j= ( + K V (x j )(K + ) d(x,x j) ) V (x j ) j= x Γ D (K + ) d(x,x j) (K + ) + K d(x,x j) K + + K 2 (K + ) + = K + <. 2. Hence, we have x Γ D R 0 (x, x) R 0 (x, x) Due to equation (9) and (0), we have (K + )2 K j= V (x j ) <. (0) R 0 (x, x) R 0 (x, x) <. x Γ. Since the difference belongs to the trace class, operators H and H have the same

27 20 a.c. spectrum. Corollary 8. Assume that for the Schrödinger operator H = + V (x) one can find the set D = {x n ; n =, 2,... } such that Γ D is the union of the disjoint finite sets ( in our terminology, D is the union of the belts). Then under the assumption of the above theorem ( n V (x n ) < ), H has no a.c. spectrum.

28 CHAPTER 5: THE REAL ANALYTIC APPROACH TO THE ABSENCE OF A.C. SPECTRUM AND LOCALIZATION In all future constructions, the belts {b l, l =, 2,... } will be selected from the assumption that for the fixed energy interval I on -axis: for all I. max z b l,z b+ (z, z 2 ) δ l, δ l 0, l () β b l There are different options to guarantee the small values of β b l (, ). For instance, assume that V (x) h l, x b l and h l is sufficiently large, then, for fixed energy interval I, i.e. V (z) 2, I, z b l, h l β b l (, ) C ( 2K h l ) tl = δ l, with t l = d( b l, b+ l ), for some constant C. The each product in the cluster expansion contains factors β b l (, ) which will be small and the resolvents of the main blocks R (l) (, ), where we have no information about the potentials. We will use here the Kolmogorov s lemma which states that R (l) (, ) for typical is not too large (see the proof of this result in [0] in the

29 22 convenient form). Proposition 9 (Kolmogorov Lemma). Let M > 0, then ( ) N α l m : F (z) = l M Here l are different real numbers, and α l are also real. l= 4 l α l M. (2) If α l > 0, l =, 2,..., N, then elementary algebraic calculations give m( : F (z) M) = 2 l α l M. If α l, l =, 2,..., N have positive terms α + l ( α + i m( : F (z) M) m : i M 2 i and negative terms α l, then ) ( α j +m : j M 2 i ) 4 l α l M. The following proposition is the central point of our approach. Proposition 0. Let H B = + V (x) with Dirichlet boundary condition: φ(x) = ( ) 0, x Γ B, where B Γ. Consider R (B) f, g, f, g l 2 (B), then in the obvious notations, ( R (B) f, g ) = f 2 g 2 k B k= (f, ψ k )(g, ψ k ) B k = (3) [(f, ψ k )/ f 2 ] [(g, ψ k )/ g 2 ]. (4) B k where B k, k =, 2,..., B are the eigenvalues of H B and ψ k are the corresponding orthonormal eigenfunctions. Due to Cauchy-Shwartz inequality and Kolmogorov s lemma, we have ( m R : ( R (B) f, g ) > M ) 4 f 2 g 2. (5) M

30 Corollary. In particular, let f = g = I b + l be the indicator function on b + l 23 and B = (l), we have m R : z,z 2 b + l (z (, z 2 ) = R (l) R (l) I b + l, I b + l ) > M + l 4 I b + l M + l 2 = 4 b+ l M +. l Similarly, m R : m R : m R : z b + l,z 2 b l+ z,z 2 b l+ R (l) z b l+,z 2 b + l where M l > 0 is a constant. R (l) (z, z 2 ) > M + l M l+ (z, z 2 ) > M l+ R (l) (z, z 2 ) > M + l M l+ 4 b + l b l+ ;(6) M + l M l+ 4 b l+ M ; (7) l+ 4 b + l b l+.(8) M + l M l+ Let s return now to the theorems, 2 giving the criteria for localization (p.p. spectrum) or absence of a.c. spectrum. Both of them contains R +iɛ (, ). Unfortunately, Kolmogorov s lemmma is not applicable for the complex-valued resolvent kernal R +iɛ (x, y), ɛ > 0. We need something else. In the case of localization, the real analytic result is simple, see [2] or [0]. Theorem 2. Assume that Q n Γ is the increasing family of the connected sets and R n, (x, y), n =, 2,... are the resolvents of the operators H n : the restrictions of H on Q n with Dirichlet boundary condition (ψ 0, x / Q n ). Assume also that for any x 0 Γ and for a.e. R, lim sup (R n, (x 0, y)) 2 c() <. n y Q n

31 24 Then lim ɛ 0 y Γ R +iɛ (x 0, y) 2 = y Γ (R +i0 (x 0, y)) 2 c() <, i.e. one can apply the theorem 2 on the localization (with appropriate randomization). The desirable result on the absence of a.c. spectrum (in real analytic terms) gives the following Theorem 3 (A.Gordon). If the operator H = + V in L 2 (Γ) has a.c. spectrum (say in the subspace generated by δ x0 ( )), then lim sup R n, (x 0, x 0 ) = n on the essential support of the a.c. component of the spectral measure of the element δ x0 ( ). This result will be published in the parallel article. Corollary 4. If, for any x 0 Γ, lim sup R n, (x 0, x 0 ) < n a.e. on some interval I, then the spectral measure of the operator H is singular on I (i.e. its a.c. component equal to 0 in each subspace, generated by δ x0 ( )).

32 CHAPTER 6: THEOREMS ON THE ABSENCE OF A.C. SPECTRUM AND LOCALIZATION In this chapter, we ll prove the major results of the paper. They will be illustrated by the examples in sections?? and??. Theorem (I). Consider the graph Γ introduced in section and the Hamiltonian H = + V (x). Assume that one can find the system of the counters b n, n =, 2,... (the belts of thickness ) such that for h n = max x b n V (x), and b n /h n 0, n. Then µ a.c. (H, f) 0. This result is the trivial corollary of the Theorem 4. In fact if h n b n 0, then one can find the sequence {n }: n h n b n < but The following example illustrates the above theorem. z b n V (x) b n h n n <. Example 5. Let Γ = S be the Sierpiński lattice and H = + V (x, ω), x S, where V (x, ω) is a system of unbounded i.i.d. random variables. Then P-a.s. µ a.c. (H) = 0. Proof. Consider the following events A n = { V (2 n i, ω) > h n, V (2 n w, ω) > h n }, where i = (, 0), w = ( 2, 3 2 ). Since events A n are independent and P (A n ) = P 2 ( V > h n ) > 0 for appropriate h n, n P (A n ) =. The second Borel- Cantelli lemma provides the existence P-a.s. of the infinitely many events A n. One

33 can apply now the above Theorem (I) to the counters containing only two points: b n = {2 n i, 2 n w}. 26 In fact, we didn t use here the specific structure of S and proved the following result. Corollary 6. If for the general graph Γ with conditions ν(x) K, one can find infinite system of the belts b n : b n C 0 < for appropriate C 0 and the i.i.d. potential V (x, ω) is unbounded(p ( V > A) = P (A) > 0 for any A). Then the Anderson Hamiltonian H = + V (x, ω) has no a.c. spectrum, P-a.s.. We ll return to the spectral theory of the self-similar fractal graphs(like Sierpiński lattice or snow flacks) in another paper to cover cases with the bounded random potentials, where again µ a.c. (H) = 0. The following result is more general than Theorem (I). Theorem (II). The condition δ l b + l b l 0, l (9) implies that Sp a.c. (H) =. Here δ l = max z b l,z 2 b + β b l (z, z 2 ). The proof of this result is the significant part of the future localization theorem. Proof. Selecting appropriate subsequence of the belts {b l }, one can assume without loss of generality that l δ l b + l b l <. Let s now apply the Borel-Cantelli lemma, which will show that the resolvent kernals R (l) (, ) are not too large.

34 Put M l = b l, M + l = b l, l =, 2,..., where α l > 0 is any sequence such α l α l that α l <. Then formulas in Corollary will have a form l 27 m ( R : > ) α l. And Borel-Cantelli lemma gives that for a.e. R and l L 0 (), R (l) (z, z 2 ) b + l. α l z,z 2 b + l Similarly, In the case z,z 2 b l+ (z, z 2 ) b l+ ; (20) α l+ R (l) (z b +, z 2 ) l b l+ ; (2) αl α l+ R (l) (z b +, z 2 ) l b l+. (22) αl α l+ R (l) z b + l,z 2 b l+ z b l+,z 2 b + l R (0) (x 0, z ) one can put z b z b (x 0, z ) R (0) b α. Firstly, assume L 0 () = 0 and choose α l = 3δ l b l b+ l. Then, due to Theorem 3, using the fact that the number of of paths [ γ] of length n is no more than 2 n,

35 28 [γ] R (x 0, x 0 ) R (0) (x 0, z ) βb (z, z 2 ) R () (z 2, z 3 ) z z 2,z 3 R (2) (z 2l 4, z 2l 3 ) z 2l 4,z 2l 3 βb 2 (z 2l 3, z 2l 2 ) R () (z 2l 2, z 2l ) z 2l 2,z 2l ( ) ( ) b b + δ b 2 b + δ α α α2 α2 βb (z 2l, z 2l ) (z z 2l R (0) where the summations are over z b, z 2 b +,..., z 2l 3 b + 2, z 2l 2 b 2, z 2l b +, z 2l b with l being the number of times the path go through the belts until it returns back to x 0. Therefore R (x 0, x 0 ) n=0 2 n 3 n = 3. Let S k = { R : L 0 () = k}, k 0. We proved that one the set S 0 : R (x 0, x 0 ) = lim N R(N) (x 0, x 0 ) and A. Gordon s Theorem 5.5 states that the a.c. component of the spectral measure ρ f (d), f = δ x0 (x) equals to 0 on S 0. Assumen that L 0 () > 0. If L 0 () = k, we introduce the new operator H N = + Ṽ where V (x) Ṽ = A k if x (m) : m k if x (m) : m < k We select constants A k so large that on b m : m < k such that b m b + m δ m α m < 3.

36 29 Then for the resolvent kernel R,k (x, x 0 ) = ( ( ) Hk I δx0(x), δ x0), we can repeat for any k the previous consideration, which gives that on the set S k { R : L 0 () = k}, there is no a.c. spectrum of H k for f = δ x0 (x). Due to Kato-Birman theorem [], the same is true for H (transition from H to H k is the finite rank perturbation of H). It proves the theorem. The last theorem in this section gives the sufficient conditions for the localization. Theorem (III). Assume that there exists the sequence of the belts {b l, l } with boundaries b ± l such that for given energy interval I, the quantities {δ l, l } introduced above satisfy the central condition of the Theorem (II): δ l b + l b l 0, l and the following series converges for a.e. I: ( n ) (n) ( δl 2 b + l b l b + n + n= l= b n+ )n +δ for some δ > 0 where (n) is the cardinality of the nth main block. Then for any x 0 Γ R (x 0, ) L 2 (Γ), for a.e. I. Corollary 7. If one can construct such belts for the random Schrödinger operator (and arbitrary x 0 Γ): H = + V (x, ω), ω (Ω, F, P ) with i.i.d. random variable V (x, ), x Γ which have bounded distribution density

37 30 f(ν), then P a.s. the spectrum of H = H(ω) is pure point. We will prove this theorem using the following strategy. Let s fix point x 0 Γ and consider the resolvent kernel R (x 0, x), I. Using the convergence of the series δ l b + l b l and the technical Lemma 8 (similar to Kolmogorov s lemma) we ll l= prove that R (x 0, ) 2 = R(x 2 0, x) < x Ω P-a.s. and a.e. for I. In this proof we will not use the existence of the distribution density p(x) for the random variables V (x, ω), but only the assumptions on δ l, and (l). In fact we ll prove that R (x 0, ) 2 < for fixed value of the potential at the reference point x 0 Γ. Using rank-one perturbation (see details in [2] or [0]), one can present R (x 0, x) as the rational function of V (x 0 ) and R (x 0, x), where R (x 0, x) is the corresponding resolvent kernel of the operator H = + Ṽ (x) where Ṽ (x) = V (x) for x x 0 and Ṽ (x 0) = 0. Since V (x 0 ) has a.c. distribution (density), the Simon-Wolf criterion [4] (H = H+V (x 0, ω)δ x0 ) gives now that spectrum of H is pure point P-a.s. on the interval I in the cyclic subspace l 2 (Γ, δ x0 ) l 2 (Γ) generated by the function f(x) = δ x0 (x). But the point x 0 is arbitrary and l 2 (Γ) is generated by the δ-functions δ x0, x 0 Γ. It completes the proof.

38 Lemma 8. Let h (n) 2 () = a b + n x (n) R(a, 2 x) for n. then 3 { } m : h (n) 2 () > M 4 (n) b + n. (23) M This Lemma is a generalization of the similar result in [0], where b + n contains a single point. In fact, for / { n,k } Here ζ n,k = a b + n h (n) 2 () = a b + n x (n) = a b + k= n (n) (n) ψ 2 n,k(a) = I b + n 2 k= ψ n,k (a)ψ n,k (x) n,k ψn,k 2 (a) (n) ( n,k ) = 2 k= 2 ζ n,k ( n,k ) 2. and {ψ n,k }, { n,k }, k =,..., (n) are the eigenfuntions and eigenvalues of the restriction of H onto the main block (n). Now one can repeat the corresponding arguments in [0]. The similar arguments work for summations over a b n+. Corollary 9. Using the Borel-Cantelli lemma, one can prove that, for any δ > 0, P-a.s. R(a, 2 x) 4 ( (n) b + n + ) b n n +δ (24) for n n 0 (ω). a b + n b n x (n) Now we can repeat the same calculation as in [0] (Theorem 2.4) or in Theorem (II) above: we assume first that we have desirable estimations for R (l) (a, b), a b+ l, b b + l or a b + l, b b l+, etc. and for R(l) (a, ) 2 2 (which are true for all l l 0 (ω) P-a.s. and a.e. in I due to Borel-Cantelli lemma). I.e. we assume that estimates (2) (23) and (24) have place for n. Consider some path [ γ] on the set of main

39 32 blocks (l), l =, 2,... which end at (n). We ll estimate first the contribution of [ γ] into R (n) (a, ) 2 2. The main contribution is given by the shortest path (0) () (n) of length n and it can be estimated as R (x 0, x) 2 n l= (δ l b + l b l ) 2 a b + n [ R (n) (a, x) ] 2. Since δ l b + l b l, the contributions of the paths of length n+, n+2,..., n+ 3 k,... are exponentially decaying in k. Summations over all paths [ γ] of the length n, n +,... and after over n complete the proof for l 0 (ω) =. If l 0 (ω), we can consider new Hamiltonian H which has very large potentials inside the first l 0 main blocks and our initial potential V (x, ω) inside blocks (n) with n > l 0. The operator H is the finite rank perturbation of H, which preserves the square integrability of the resolvent, see [2]. But for H we have desirable estimations for all l. The idea of transition using finite rank perturbation from the general l 0 to l 0 = is the same as in the proof of the absence of the a.c. spectrum (Theorem (I), (II)).

40 CHAPTER 7: EXAMPLES We will illustrate the general Theorem (III) by several examples. In all these examples, the belts will be relatively short. The belt factors in these examples will be compensated by large values of the potential on the belts. Example 20 (Localization on Sierpiński lattice S ). Let s start from the fractal (nested) lattice and consider as a typical example the Sierpiński Lattice S. Let S n be the part of S with vertices 0, 2 n i, and 2 n w. The volume of S n is given as S n = 3n See fig. 2 Consider the Anderson Hamiltonian H = + V ( x, ω), where x S and V (, ω) are unbounded i.i.d. random variables with bounded density function f(x) on R. Consider for fixed A the sequence of independent events B A,n = { V (2 n i, ω) > A, V (2 n w, ω) > A}, where i = (, 0) and w = (/2, 3/2). Then P (B A,n ) = ( 2 p 2 (A) = f(x)dx). Let τ A be the moment of the first occurrence of B A,n x >A in the sequence B A,0, B A,,.... Then τ A has geometric distribution: P (τ A = k) = ( p 2 (A)) k p 2 (A) for k =, 2,..., with Eτ A = p 2 (A)τ A law Exp() as A. p 2 (A). It is easy to see that Consider the increasing sequence {A n = n} : n and the moments τ n. Since P ( p 2 (n)τ n > ( + ɛ) ln n ) n n we have ( due to Borel- Cantelli lemma) P-a.s. τ n c 0 n +ɛ < for some constant c 0 and any ɛ > 0, ( + ɛ) ln n, for n n p 2 0 (ω). (n)

41 The successive belts b n, n contains the pairs of the points {2 τ An i, 2 τ An w}. Of ( course, we have (n) c 3 τn c exp [( + ɛ) ln 3 ln n ) < and for fixed energy p 2 (A) interval I, β b k c 2(I) n, where c, c 2 are some constants. Assume that P { V ( ) > A} = p(a) c 3 A θ Then Theorem (III) provides the P-a.s. localization certenly if ( n n k= β b k (since ( b ) (n) + n + b n+ 34 for any A > 0, where c 3 is a constant. ) 2 (n) exp ( c 4 n 2n ln n + c 5 n 2θ ln n ) n 4 (n) ). The last series converges if θ < 2. Theorem 2. Condition P { V ( ) > A} = p(a) localization on S. c A θ, θ < 2 is sufficient for P-a.s. The same proof works for all nested fractal lattices. Let s stress that we didn t use here the fundamental properties of the self-similarity of S. The spectral analysis of the Laplacian on S can provide much better localization results and cover the case of the light tails of V (, ω). We ll return to this subject in other publications and prove the localization for cases when p(a) C A for θ any θ > 0. Example 22. Consider the Quasi- dimensional tree as shown in figure 2, denoted by T. The set of vertices is { x = (x, x 2 ) : x, x 2 are nonnegative integers} {(, 0)}. Consider the Anderson Hamiltonian H = + V ( x, ω) on T, where V (, ω) is are i.i.d. random variables with density f(x) such that P (V ( x; w) > A) = A f(x)dx = p(a) > 0 for all A R. For fixed energy interval I, let s select a constant A such that A 2, I and

42 introduce the following points on x axis and on the vertical lines {(x, y) : y > 0} for positive integers x. Put 35 τ = min{x > 0 : V (x, 0, ω) > A, } τ 2 = min{x : V (x, 0, ω) > A, V (x +, 0, ω) > A}... τ n = min{x > τ n : V (x, 0, ω) > A,..., V (x + n, 0, ω) > A}.... Similarly, for fixed x, on the vertical line {(x, y) : y > 0}, we define τ x, = min{y > 0 : V (x, y, ω) > A, } τ x,2 = min{y > τ x, : V (x, y, ω) > A, V (x, y +, ω) > A}... τ x,n = min{y > τ x,n : V (x, y, ω) > A,..., V (x, y + n, ω) > A}... The random variables τ, τ 2 τ,..., τ n τ n,... ; τ x,, τ x,2 τ x,,..., τ x,n τ x,n,..., x = 0,, 2,... are independent. As easy to see, τ n τ n or τ x,n τ x,n are majorated by the random variable nθn or nθx,n, where θn and θx,n are geometrically distributed with parameter p n (A). Like in the previous example p n (A)θn Exp(). and Borel- Cantelli lemma gives P-a.s. θ n ( + ɛ) ln n, n n p n 0 (ω) (A)

43 36 i.e. τ n τ n ( + ɛ)n ln n, n n p n 0 (ω) (A) The same calculations show that p n (A)θ x,n ( + ɛ)(ln n + ln x) except finitely many pairs (x, n), i.e. τ x,n ( + ɛ) ln(nx).x + n n p n (ω). (A) The belt b n consists of the points {(x, 0), τ n x τ n + n } on the x axis and for any fixed x the points {(x, y), τ x,n y τ x,n + n }. As a result, (n) ( + ɛ)n ln n p n (A) ( ( + ɛ)n ln n p n (A) ) (+ɛ)n ln n p n (A) c(a)n3 ln n p 2n (A) for some c(a), ϑ(a) > 0. Also we have β bn ( 2) n and some ϑ > 0. Applying the general result of Theorem (III), we ll get ( n k= c(a)n 3 ln ne ϑ(a)n β b k ) 2 e ϑn2, for Theorem 23. For the Anderson Hamiltonian on the graph T (see figure 2)with boundary condition ψ(, 0) = 0, where V ( x, ω) are i.i.d. random variables with bounded distribution density f(ν) such that f(ν)dν = p(a) > 0 for any A > 0, the ν >A spectrum of H is pure point with probability. Remark 2. One can prove that the spectrum of the pure Laplacian on the graph T is a.c.. Remark 3. The Hausdorff dimension of the graph T equals 2: it is simply the lattice Z 2 after removing some edges.

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