Green's Function Estimates for Lattice Schrödinger Operators and Applications. (AM-158) 9781400837144

Table of contents :
Contents
Acknowledgment
Chapter 1. Introduction
Chapter 2. Transfer Matrix and Lyapounov Exponent
Chapter 3. Herman’s Subharmonicity Method
Chapter 4. Estimates on Subharmonic Functions
Chapter 5. LDT for Shift Model
Chapter 6. Avalanche Principle in SL2(R)
Chapter 7. Consequences for LyapounovExponent, IDS, and Green’s Function
Chapter 8. Refinements
Chapter 9. Some Facts about Semialgebraic Sets
Chapter 10. Localization
Chapter 11. Generalization to Certain Long-Range Models
Chapter 12. Lyapounov Exponent and Spectrum
Chapter 13. Point Spectrum in Multifrequency Models at Small Disorder
Chapter 14. A Matrix-Valued Cartan-Type Theorem
Chapter 15. Application to Jacobi Matrices Associated with Skew Shifts
Chapter 16. Application to the Kicked Rotor Problem
Chapter 17. Quasi-Periodic Localization on the Z^d-lattice (d > 1)
Chapter 18. An Approach to Melnikov’s Theorem on Persistency of Non-resonant Lower Dimension Tori
Chapter 19. Application to the Construction of Quasi-Periodic Solutions of Nonlinear Schrödinger Equations
Chapter 20. Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations
Appendix

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Green’s Function Estimates for Lattice Schrödinger Operators and Applications

J. Bourgain

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

Copyright © 2005 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Control Number: 2004104492 ISBN: 691-12097-8 (hardcover); 0-691-12098-6 (paper) British Library Cataloging-in-Publication Data is available This book has been composed in LaTeX The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper. pup.princeton.edu Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Contents

Acknowledgment

v

Chapter 1. Introduction

1

Chapter 2. Transfer Matrix and Lyapounov Exponent

11

Chapter 3. Herman’s Subharmonicity Method

15

Chapter 4. Estimates on Subharmonic Functions

19

Chapter 5. LDT for Shift Model

25

Chapter 6. Avalanche Principle in SL2 (R)

29

Chapter 7. Consequences for Lyapounov Exponent, IDS, and Green’s Func31 tion Chapter 8. Refinements

39

Chapter 9. Some Facts about Semialgebraic Sets

49

Chapter 10. Localization

55

Chapter 11. Generalization to Certain Long-Range Models

65

Chapter 12. Lyapounov Exponent and Spectrum

75

Chapter 13. Point Spectrum in Multifrequency Models at Small Disorder

87

Chapter 14. A Matrix-Valued Cartan-Type Theorem

97

Chapter 15. Application to Jacobi Matrices Associated with Skew Shifts

105

Chapter 16. Application to the Kicked Rotor Problem

117

Chapter 17. Quasi-Periodic Localization on the Zd -lattice (d > 1)

123

Chapter 18. An Approach to Melnikov’s Theorem on Persistency of Non133 resonant Lower Dimension Tori

iv

CONTENTS

Chapter 19. Application to the Construction of Quasi-Periodic Solutions of 143 Nonlinear Schrödinger Equations Chapter 20. Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations 159 Appendix

169

Acknowledgment The author is grateful to W. Schlag, Silvius Klein, and Ya-Ju Tsai for reading and comments on the initial versions of this manuscript. The author would also like to acknowledge the National Science Foundation, which provided partial support for the research for this book with grant DMS 0322370.

Green’s Function Estimates for Lattice Schrödinger Operators and Applications

Chapter One Introduction We will consider infinite matrices indexed by Z (or Zb ) associated to a dynamical system in the sense that   H = H(x)m,n m,n∈Z satisfies H(x)m+1,n+1 = H(T x)m,n where x ∈ Ω, and T is an ergodic measure-preserving transformation of Ω. Typical settings considered here are Ω=T Ω = Td Ω = T2 Ω = T2

Tx = x + ω (1 − frequency shift) Tx = x + ω (d − frequency shift) T x = (x1 + x2 , x2 + ω) (skewshift) T x = Ax, where A ∈ SL2 (Z), hyperbolic

Thus H(x)m,n = φm−n (T m x)

(1.0)

where the φk are functions on Ω. We will usually assume that H(x) is self-adjoint, although many parts of our analysis are independent of this fact. Define HN = R[1,N ] HR[1,N ] where R[1,N ] = coordinate restriction to [1, N ] ⊂ Z, and the associated Green’s functions are GN (E) = (HN − E)−1 (if HN − E is an invertible N × N matrix). One of our concerns will be to obtain a ‘good bound’ on GN (E, x), except for x in a “small” exceptional set. A typical statement would be the following: GN (E, x) < eN

1−δ

(1.1)

and |GN (E, x)(m, n)| < e−c|m−n| for |m − n| >

N 10

(1.2)

for all x outside a set of measure < e−N . Here δ, σ > 0 are some constants. The exceptional set in x does depend on E, of course. Such estimates are of importance in the following problems, for instance. σ

2

CHAPTER 1

1. Spectral problems for lattice Schrödinger operators Description of the spectrum Spec H(x) and eigenstates of H(x) (i.e., point spectrum, continuous (absolutely continuous or singular continuous) spectrum, localization, extended states, etc.) 2. Long-time behavior of linear time-dependent Schrödinger operators i

∂u + Δu + V (x, t)u = 0 ∂t

(1.3)

The spatial variable x ∈ Td (i.e., periodic bc). The potential V depends on time. It is well known that if V is periodic in time (say, 1-periodic), we are led to study the monodromy operator W u(t) = u(t + 1) (which is unitary). Again, the nature of spectrum and localization of eigenfunctions are key issues. A well known example is the so-called kicked rotor problem    ∂u ∂ 2u ∂u i + a 2 + ib + κ cos x δ(t − n) u = 0 (1.4) ∂t ∂x ∂x n∈Z

involving periodic “kicks” in time introduced as a model in quantum chaos. Here V is discontinuous in time. We assume V real. We will also assume V (·, t) smooth in x ∈ Td for all time. By the reality of V , there is conservation of the L2 -norm. If u0 = u(0) ∈ H s (Td ), then u(t) ∈ H s for all time Problem. Possible growth of u(t)H s . Remark 1. It turns out that in (1.4) with typical values of a, b there is almostperiodicity in the following sense: Assume u0 sufficiently smooth (depending on s). Then u(t) is almost as periodic as an H s -valued function and, in particular, supt u(t)H s < ∞. Remark 2. If in (1.3) we take V also to be t-periodic, u(t) is well known to be almost periodic in time as an L2 -valued function. But there are examples where V is smooth in x and t and such that for some smooth initial data u 0 sup u(t)H s = ∞ for all s > 0 t

3. KAM-theory via the Nash-Moser method We refer here to a method developed by W. Craig, G. Wayne, and myself to construct quasi-periodic solutions of nonlinear Hamiltonian PDEs. This approach was used originally as a substitute of the usual KAM-scheme (as used in this context by S. Kuksin) in situations involving multiplicities or near-multiplicities of normal frequencies. These always appear, except in 1D problems with Dirichlet boundary

3

INTRODUCTION

conditions. It was realized later that this technique is also of interest in the “classical context” involving finite-dimensional phase space (leading, for instance, to a Melnikov-type result with the “right” nonresonance assumptions) and applies in certain non-Hamiltonian settings. If we follow a Newton-type iteration scheme, the basic difficulty is the inversion of nondiagonal operators obtained by linearizing the (nonlinear) PDE. Consider, for instance, the Schrödinger case iut + Δu + εF (u, u ¯) = 0

(1.5)

The linearized operator expressed in Fourier modes then becomes T = D + εS where D is diagonal with diagonal elements of the form Dk,n = k.ω + μn = k.ω + |n|2 + 0(1) (k ∈ Zb , n ∈ Zd )

(1.6)

and S is a Toeplitz-type matrix with (very) smooth symbol, i.e.,   ˆ − k  , n − n ) S (k, n), (k  , n ) = ϕ(k where ϕ(ξ) ˆ decays rapidly for |ξ| → ∞. In (1.6), b = dimension of invariant tori, and ω ∈ Rb is the frequency vector. The matrix T is finite (depending on the iteration step), and we seek appropriate bounds on T −1 . The problem again involves small-divisor issues and is treated by multiscale analysis. Returning to H(x), one important special case is given by H(x) = λ.v(T n x)δnn + Δ

(1.7)

where Δ is the usual lattice Laplacian Δ(n, n ) = 1 if |n − n | = 1 = 0 otherwise Letting v(x) = cos x on T, T x = x + ω = shift, we obtain the Almost Mathieu operator Hλ (x) = λ cos(x + nω) + Δ

(1.8)

introduced by Peierls and Hofstadter in the study of a Bloch electron in a magnetic field and studied extensively afterwards by many authors. For (1.8), there is basically a complete understanding of the nature of the spectrum. Assume that ω satisfies a diophantine condition dist (kω, 2πZ) = k.ω > c|k|−C for k ∈ Z\{0} Then, for a.e. x, (i) λ > 2: H(x) has p.p. spectrum (ii) λ = 2: H(x) has purely s.c. spectrum (iii) λ < 2: H(x) has purely a.c. spectrum

4

CHAPTER 1

Thus there is a phase transition at λ = 2. This model has a special and remarkable self-duality property (wrt Fourier transform) cos → 12 Δ Δ → 2 cos observed and exploited first by Aubry. One of its implications is that Spec Hλ = Spec H λ4 (referring to the “topological spectrum” that is independent of x). In more general situations involving shifts, λv(x + nω)δnn + Δ

(1.9)

with v real analytic on T , a rough picture is the following: λ large: p.p. spectrum with Anderson localization λ small: purely a.c. spectrum λ intermediate: possible coexistence of different spectral types Recall that Anderson localization means the following: Assume ψ an extended state, i.e., d

< |n|C Hψ = Eψ and |ψn | ∼

Then ψ ∈ 2 and |ψn | < e−c|n| for |n| → ∞ (in particular, E is an eigenvalue). Related to possible coexistence of different spectral types (in various energy regions), one may prove the following: Consider H = (λ cos nω1 + τ cos nω2 )δnn + Δ

(1.10)

where λ < 2, and τ is small. Then, for ω = (ω1 , ω1 ) in a set of positive measure, H has both point spectrum and a.c. spectrum. Remark. If in (1.9) we replace the shift by the skew shift, one expects a different spectral behavior with localization for all λ > 0 (as is the case of a random potential). This problem is open at this time. It is known that for all λ > 0 and ω in a set of positive measure   n(n − 1) H = λ cos ω δnn + Δ 2 has some p.p. spectrum. This text originates from lectures given at the University of California, Irvine, in 2000 and UCLA in 2001. The first 17 chapters deal mainly with localization problems for quasi-periodic lattice Schrödinger operators. Part of this material is borrowed from the original research papers. However, we did revise the proofs in order to present them in a concise form with emphasis on the key analytical points. The main interest, independent of style, is that we give an overview of a large body of

INTRODUCTION

5

research presently scattered in the literature. The results in Chapter 8 on regularity properties of the Lyapounov exponent and Integrated Density of states (IDS) are new. They refine the work from [G-S] described in Chapter 7. (Nonperturbative quasi-periodic localization is discussed in Chapter 10. We follow the paper [BG] but also treat the general multifrequency case (in 1D). In [B-G], only the case of two frequencies was considered. Our presentation here uses the full theory of semialgebraic sets and in particular the Yomdin-Gromov uniformization theorem. This material is discussed in Chapter 9. Chapters 18, 19, and 20 deal with the problem of constructing quasi-periodic solutions for infinite-dimensional Hamiltonian systems given by nonlinear Schrödinger (NLS) or nonlinear wave equations (NLW). Earlier research, mainly due to C. Wayne, S. Kuksin, W. Craig, and myself (see [C] for a review), left open a number of problems. Roughly, only 1D models and the 2D NLS could be treated. In this work we develop a method to deal with this problem in general. Thus we consider NLS and NLW (with periodic boundary conditions) given by a smooth Hamiltonian perturbation of a linear equation with parameters and proof persistency of a large family of smooth quasi-periodic solutions of the linear equation. This is achieved in arbitrary dimension. Compared with earlier works, such as [C-W] and [B1], we do rely here on more powerful methods to control Green’s functions. These methods were developed initially to study quasi-periodic localization problems. Thus the material in Chapters 18 to 20 is also new. We want to emphasize that it is our only purpose here to convey a number of recent developments in the general area of quasi-periodic localization and the many remaining problems. This is an ongoing area of research, and our understanding of most issues is still far from fully satisfactory. The material discussed, moreover, covers only a portion of these developments (for instance, we don’t discuss at all renormalization methods, as initiated by B. Hellfer and J. Sjostrand). We have largely ignored the historical perspective. Nevertheless, it should be pointed out that this field to a large extent owes its existence to the seminal work of Y. Sinai and his collaborators (in particular, the papers [Si], [C-S], and [D-S]), as well as the paper [F-S-W] by Frohlich, Spencer, and Wittwer. One of the significant differences, however, between these works (and some later developments such as [E]) and ours on the technological side is the fact that we don’t rely on eigenvalue parametrization methods, which seem, in particular, very hard to pursue in multidimensional problems (such as considered in [B-G-S], for instance). It turns out that, as mentioned earlier, lots of the analysis is independent of self-adjointness and has potential applications to non-self-adjoint problems. We rely heavily in both perturbative and nonperturbative settings on methods from subharmonic function theory and the theory of semianalytic sets, which somehow turn out to be more “robust” than eigenvalue techniques (the results obtained are a bit weaker in the sense that “good” frequencies are not always characterized by diophantine conditions, as in [Si], [F-S-W], [E], or [J]). Jitomirskaya’s paper [J] certainly underlies much of this recent research. Besides settling the spectral picture for the Almost Mathieu operator and the phase transition mentioned earlier, it initiated the nonperturbative approach with emphasis on the Lyapounov exponent and transfer matrix. Some parts of the analysis were restricted to the cosine potential, and the extension to gen-

6

CHAPTER 1

eral polynomial or real analytic potentials (see [B-G]) lies at the root of the material presented in these notes. Next, a bit more detailed discussion of the content of the different chapters. Chapters 2 through 11 are closely related to the papers [B-G] and [G-S] on nonperturbative localization for quasi-periodic lattice Schrödinger operators of the form Hx = λv(x + nω) + Δ

(1.11)

where v is a real analytic potential on Td (d = 1 or d > 1), and Δ denotes the lattice Laplacian on Z. We are mainly concerned with the issues of pure point spectrum, Anderson localization, dynamical localization, and regularity properties of the IDS. A key ingredient is the positivity of the Lyapounov exponent for sufficiently large λ. The results are nonperturbative in the sense that the condition λ > λ0 (v) depends on v only and not on the arithmetical properties of the rotation vector ω (provided we assume ω to satisfy some diophantine condition). Here and throughout this exposition, extensive use is made of subharmonic function techniques and the theory of semialgebraic sets. A summary of certain basic results in semialgebraic set theory appears in Chapter 9. The basic localization theorem is proven in Chapter 10, and some extensions of the method to more general operators are given in Chapter 11. In Chapter 12 we recall some elements from Kotani’s theory for later use. But this is far from a complete treatment of this topic, and several other results and aspects are not mentioned. In Chapter 13 we exhibit point spectrum in certain two-frequency models of the form (0.11) with small λ. This fact shows that, contrary to the localization theory, the nonperturbative results on absolutely continuous spectrum, as obtained in [B-J] for one-frequency models, fail in the multifrequency case. Equivalently, invoking the Aubry duality, the quasi-periodic localization results on the Z2 -lattice (as discussed in Chapter 17) are only perturbative. In Chapter 14 we develop a general perturbative method to control Green’s functions of certain lattice Schrödinger operators. The main result is in some way an “analogue” of Cartan’s theorem in analytic function theory for holomorphic matrixvalued functions. This approach has a wide range of applications. First, it allows us to control Green’s functions for general Jacobi operators of the form (1.0) associated to a dynamical system given by a skew shift (Chapter 15). As an application, we prove the almost periodicity of smooth solutions of the kicked rotor equation (1.4) with small κ and typical parameter values a, b (Chapter 16). Next, an extension of Chapter 14 to a 2D setting permits us to establish Anderson localization for operators of the form (1.11) on the Z2 -lattice. The statement is perturbative, i.e., λ > λ0 (v, ω). However, as indicated earlier, a nonperturbative result may be false in this situation. In fact, considering the multifrequency generalizations of the Almost-Mathieu operator   Hx = λ cos(x1 + nω1 ) + cos(x2 + nω2 ) + Δ (on Z) (1.12) and its “dual” θ = cos(θ + n1 ω1 + n2 ω2 ) + λ Δ H 4

(on Z2 )

(1.13)

7

INTRODUCTION

it turns out that for arbitrary λ > 0, there is a set of frequencies Ω = Ωλ ⊂ T2 of small but positive measure such that for ω ∈ Ω and x in a set of positive measure, we have   mes Hx > 0 pp

θ has (in fact, there may be coexistence of different spectral types here). Hence H 2  true (i.e., not  ) extended states for almost all θ. (Z -operators of the form (1.13) were first studied in [C-D].) Finally, the method from Chapter 14 enable us to treat KAM-type problems via the Lyapounov-Schmidt approach (see [C-W]) in a number of situations that, due to large sets of resonances, seemed untractable previously. (Typical issues left open here from the earlier works are the NLS in space dimension D ≥ 3 and the NLW in space dimension D ≥ 2). In Chapter 18 we give a new proof of Melnikov’s theorem on persistency of bdimensional tori in (finite-dimensional) phase space of dimension > 2b (for Hamiltonian perturbations of a linear system, assuming the Hamiltonian given by a polynomial.) The spirit of the argument is closely related to earlier discussion on perturbative localization. In particular, semialgebraic set theory is used again to restrict the parameter space. In Chapters 19 and 20 we then apply this scheme to obtain quasi-periodic solutions for nonlinear PDE (with periodic bc), thus involving an infinite-dimensional phase space. Chapter 19 deals with NLS and Chapter 20 with NLW. Compared with the finite-dimensional phase space setting discussed in Chapter 18, there are some additional difficulties (due to large sets of resonant normal modes). But the method is sufficiently robust to deal with them. An additional ingredient involved here is a “separated cluster structure” for the near-resonant sets (noticed first by T. Spencer in a 2D-Schrödinger context). As mentioned earlier, results from Chapters 18 to 20 treat only perturbations of linear systems with parameters. Starting from a genuine nonlinear problem, this format may often be reached through the theory of normal forms and amplitudefrequency modulation (see [K-P] and [B2]). This is a different aspect of the general problem, however, that is not addressed here. In the Appendix we consider lattice Schrödinger operators associated to strongly mixing dynamical systems. We mainly summarize results from [C-S] and [B-S] based on the Figotin-Pastur approach. So far, this method to evaluate Lyapounov exponent has succeeded only in a strongly mixing context.

8

CHAPTER 1

References [B1] J. Bourgain. Quasi-periodic solutions of Hamiltonian perturbations of 2Dlinear Schrödinger equations, Annals of Math. 148 (1998), 363–349. [B2] J. Bourgain. Construction of periodic solutions of nonlinear wave equations in higher dimension, GAFA 5 (1995), 629–639. [G-B] J. Bourgain, M. Goldstein. On non-perturbative localization with quasiperiodic potential, Annals of Math. (2) 152(3) (2000), 835–879. [BGS] J. Bourgain, M. Goldstein, W. Schlag. Anderson localization for Schrödinger operators on Z2 with quasi-periodic potential, Acta Math. 188 (2002), 41–86. [B-J] J. Bourgain, S. Jitomirskaya. Nonperturbative absolutely continuous spectrum for 1D quasi-periodic operators, preprint 2000. [B-S] J. Bourgain, W. Schlag. Anderson localization for Schrödinger operators on Z with strongly mixing potentials, CMP 215 (2000), 143–175. [C-D] V. Chulaevsky, E. Dinaburg. Methods of KAM theory for long-range quasi-periodic operators on Zn . Pure point spectrum, CMP 153(3) (1993), 539–557. [C-S] V. Chulaevsky, T. Spencer. Positive Lyapounov exponents for a class of deterministic potentials, CMP 168 (1995), 455–466. [C-Si] V. Chulaevsky, Y. Sinai. Anderson localization for multifrequency quasiperiodic potentials in one dimension, CMP 125 (1989), 91–121. [C] W. Craig. Problemes de petits diviseurs dans les Equations aux dériveés partielles, Panoramas et Synthèses, 9, SMF, Paris, 2000. [C-W] W. Craig, C. Wayne. Newton’s method and periodic solutions of nonlinear wave equations, CPAM 46 (1993), 1409–1501. [D-S] E.I. Dinaburg, Y. Sinai. Methods of KAM-theory for long-range quasiperiodic operators on Zν . Pure point spectrum, CMP 153(3) (1993), 559– 577. [E] L.H. Eliasson. Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum, Acta Math. 179 (1997), 153–196. [F-S-W] J. Fröhlich, T. Spencer, P. Wittwer. Localization for a class of one dimensional quasi-periodic Schrödinger operators, Comm. Math. Phys. 132 (1990), 5–25.

INTRODUCTION

9

[G-S] M. Goldstein, W. Schlag. Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions, Annals of Math. 154 (2001) 155–203. [J] S. Jitomirskaya. Metal-insulator transition for the Almost Mathieu operator, Annals of Math. (2) 150(3) (1999), 1159–1175. [K-P] S. Kuksin, J. Pöschel. Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Annals of Math. 143 (1996), 149–179. [Si] Ya. G. Sinai. Anderson localization for one-dimensional difference Schrödinger operator with quasi-periodic potential, J. Statist. Phys. 46 (1987), 861–909.

Chapter Two Transfer Matrix and Lyapounov Exponent Consider 1D lattice Schrödinger operators of the form H = v(T j x)δjj  + Δ Assume that ψ = (ψj )j∈Z is a sequence satisfying Hψ = Eψ Then



ψn+1 ψn



 = Mn (E)

where Mn (E) = Mn (E, x) =

ψ1 ψ0



1 

v(T j x) − E 1

j=n

−1 0



is the transfer (or fundamental) matrix. Define further  1 LN (E) = log MN (x, E)dx N

(2.1)

(2.2)

and L(E) = lim LN (E) N →∞

= Lyapounov exponent

(2.3)

Observe that by submultiplicativity Mn1 +n2 (x, E) ≤ Mn2 (T n1 x, E).Mn1 (x, E) hence Ln1 +n2 (E) ≤ L(E)

n1 n1 +n2 Ln1 (E)

+

n2 n1 +n2 Ln2 (E)

= lim Ln (E) exists n→∞

and by Kingman’s ergodic theorem (assuming T ergodic) (see [K]) 1 log Mn (x, E) x a.e. n There is the following relation between Mn (E) and determinants     det(H  n (x) − E)  − det Hn−1 (T x) − E  Mn (x, E) = − det Hn−2 (T x) − E det Hn−1 (x) − E L(E) = lim

n→∞

(2.4)

12

CHAPTER 2

Define the integrated density of states as  1  # ] − ∞, E] ∩ Spec Hn (x) x a.e. N (E) = lim n→∞ n The relation to the Lyapounov exponent is expressed by the Thouless formula (see [S], for instance)  L(E) = log |E − E  |dN (E  ) (2.5) The convergence N1 log MN (x, E) → L(E) can be made more precise in certain cases by exploiting the specific structure (in particular, the transformation T ). In what follows, an important role will be played by large deviation theorems (LDT) of the form d 1 mes[x ∈ T log MN (x, E) − LN (E) > κ] < δ(N, κ) (2.6) N →∞ 0 (fixing κ > 0). where δ(N, κ) N→ This bound δ(N, κ) will usually be exponential in N ; thus σ δ(N, κ) < e−N (σ > 0) for N large enough. Such estimates are of particular interest in estimating Green’s functions by Cramer’s rule. One has for 1 ≤ n1 ≤ n2 ≤ N , by (2.4)  −1 |GN (E, x)(n1 , n2 )| = | HN (x) − E |(n1 , n2 )|

=

| det[Hn1 −1 (x)−E]| | det[HN −n2 (T n2 x)−E]| | det[HN (x)−E]|

≤

Mn1 (x,E) MN −n2 (T | det[HN (x)−E]|

n2

(2.7)

x,E)

Assume that LN0 (E) < L(E) + κ and

1 log Mn (y, E) − Ln (E) < κ n for N0 ≤ n ≤ N and y ∈ {x, T x, . . . , T N x}. Then e(N −|n1 −n2 |)L(E)+2κN +0(N0) (2.7) < (2.8) | det[HN (x) − E]| Returning to (2.4), if we allow replacement of N by N − 1 or N − 2 and x by T x, we may replace the denominator in (2.8) by MN (x, E) > eN L(E)−2κN . Thus we obtain GΛ (E, x)(n1 , n2 ) < e−L(E)|n1 −n2 |+0(κN +N0 ) (2.9) where Λ is one of the intervals [1, N ], [1, N − 1], [2, N ], [2, N − 1] It is clear from (2.9) that positivity of the Lyapounov exponent L(E) > c > 0 is important to get decay estimates on the Green’s function. A major advantage of this technique is that it may provide nonperturbative results.

TRANSFER MATRIX AND LYAPOUNOV EXPONENT

13

References [K] J.F.C. Kingman. The ergodic theory of subadditive stochastic processes, J. Royal Statist. Soc. B30 (1968). [S] J. Avron, B. Simon. Almost periodic Schrödinger operators II. The integrated density of states, Duke Math. J. 50 (1) (1983), 369–391.

Chapter Three Herman’s Subharmonicity Method There is a particularly simple method to obtain lower bounds on L(E) in case v(x) is a trigonometric polynomial. The argument is based on Jensen’s inequality. We consider the example v(x) = cos x. Proposition 3.1. Consider H(x) = λ cos(nω + x)δnn + Δ

Then L(E) ≥ log Proof. Write LN (E)

λ 2

(3.2)

 λ cos(θ + jω) − E −1 dθ log  = 1 0 N   λ −ijω 1

 + λ2 eijω z 2 − Ez −z 2e = N1 |z|=1 log   z 0 N N  λ −ijω 0 2e ≥ N1 log   (by Jensen) 0 0 = log λ2 . 1 N

1 



Remarks. 1. The argument clearly generalizes to trigonometric polynomials. 2. For v(x) = cos x, L(E) ≥ log λ2 is optimal as energy-independent lower bound It follows in particular that L(E) > c > 0 for all E if λ > 2 (which is the regime of p.p. spectrum and Anderson localization). If v is given by a real analytic function on T, i.e.,  v(θ) = vˆ(k)e2πikθ |ˆ v (k)| < e−ρ|k| k∈Z

then it is still possible to obtain a lower bound by subharmonicity. Proposition 3.3. Let H(x) = λv(nω + x)δnn + Δ with v as above, v nonconstant. Then, for λ > λ0 , 1 L(E) > log λ 2 where λ0 = λ0 (v). This result is due to Sorets-Spencer [S-S]. Observe that in this issue, approximation of v by trigonometric polynomials and use of Herman’s argument fails unless additional assumptions on v are made.

16

CHAPTER 3

Complexify v to the strip |Im z|
0, there is ε > 0 such that inf sup inf |v(x + iy) − E1 | > ε E1

δ 2 ε x∈R λ Define λ0 = 10ε−10 and take λ > λ0 . It follows from (3.7) that     1 1 | , Mn (iy0 , E) ≥ | Mn (iy0 , E) 0 0 > (λε − 1)n Hence u(iy0 ) > log(λε − 1) (3.8) ρ Denote μ ∈ M([y = 0] ∪ [y = 10 ]), the harmonic measure of y0 in the strip ρ 0 ≤ y ≤ 10

17

HERMAN’S SUBHARMONICITY METHOD .... ..... .. ..

ρ/10

y

y0 −

........... .......

0

x

Clearly, ρ ] < μ[y = 10 dμ < dx y=0

10y0 ρ


log λ δ 0 2 A better estimate is obtained by writing  y0 1 u(x + a) 2 dx > log λ 2 x + y0 2 R with a ∈ R arbitrary; see (3.7). Integrating in a ∈ [0, 1] thus implies  1 1 u(x)dx > log λ (λ > λ0 ) 2 0 This proves Proposition 3.3. Remarks. 1. Both Herman’s [H] and Soretz-Spencer’s [S-S] arguments are independent of diophantine assumptions on ω. 2. For v = v(x) real analytic on Td , d > 1, and considering the multifrequency shift by ω, Proposition 3.3 holds if we make a diophantine assumption on ω (see [B-G] and [G-S] in Chapter 1). The exact analogue of Proposition 3.3 for d > 1 as a uniform minoration with no dependence on ω was proven more recently in [Bo]. 3. A natural question is whether a Soretz-Spencer type theorem holds in Gevrey class. The key point in the proof of Proposition 3.3 is to avoid the set [|v| ≈ 0] by complexification. This procedure does not work for d > 1 unless again additional assumptions on v are made. The argument in [Bo] uses diophantine considerations, although the final conclusion is independent of them.

18

CHAPTER 3

References [Bo] J. Bourgain. Positivity and continuity of the Lyapounov exponent for shifts on Td with arbitrary frequency vector and real analytic potential, preprint 2002. [H] M. Herman. Une methode pour minorer les exposants de Lyapounov et quelques examples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2, Comment. Math. Helv. 58 (1983), 453–562. [S-S] E. Soretz, T. Spencer. Positive Lyapounov exponents for Schrödinger operators with quasi-periodic potentials, CMP 142(3) (1991), 543–566.

Chapter Four Estimates on Subharmonic Functions The material presented in this chapter appears in [B-G], [G-S] (see Chapter 1), and [B-G-S] with slightly different formulations and proofs. Assume u = u(x) 1-periodic with subharmonic extension u =u (z) to the strip |Im z| < 1 satisfying |u| ≤ 1, | u| ≤ B

...... ... ..

(4.1)

y

1

D ...... ...........

−1

1

1 2

− 12

x

−1

Apply Riesz representation on D as above. Thus in particular for |x| ≤ 34 ,  u(x) = log |x − w|μ(dw) + h(x)

(4.2)

where dμ = Δ u dw Hence μ is a positive measure on D, and h is harmonic on D. By (4.1), < B μ ∼ < B for |x| < |∂x(α) h| ∼

Denote

(4.3) 3 4

 log |x − w|μ(dw)

v(x) = D

(4.4)

20

CHAPTER 4

Hence ∂x v

=

x−Re w μ(dw) (x−Re w)2 +(Im w)2

(4.5)

= H[ν] (H = Hilbert-transform) where ν ≥ 0 is the measure on R given by  dν |Im w| = μ(dw) dx (x − Re w)2 + (Im w)2 Hence < B ν ≤ μ ∼ < Corollary 4.7. |ˆ u(k)| ∼

(4.6)

B |k| .

   Proof. Take smooth η, supp η ⊂ − 34 , 34 , j∈Z η(x + j) = 1. Then

u ˆ(k) = R u(x)η(x)e−2πikx dx |ˆ u(k)| ∼

1  |k| |∂x (uη)(k)|

1  |k| (|∂x (vη)(k)| < B ∼ |k|

≤

+ |∂ x (hη)(k)|)

by (4.4), (4.5), and (4.6). Corollary 4.8. Assume, moreover, that mes[x ∈ T |u(x) − u| > ε0 ] < ε1

Then uBMO(T) ≤ C(ε0 +

 ε1 B)

(4.9)

Proof. We may assume u = 0. From the hypothesis u = u0 + u1 u0 ∞ ≤ ε0 , u1 1 < ε1 Let

 ε∼

ε1 B

and denote Pε (t) = 1ε P ( εt ), with P ≥ 0 compactly supported, Since u = v + h = u0 + u1 , write

u = (u − Pε u) + Pε u0 + Pε u1 = (v − Pε v) + (h − Pε h) + Pε u0 + Pε u1 where < εB by (4.4) h − Pε h∞ ∼ Pε u0 ∞ < ε0 ε1 < Pε u1 ∞ ∼ ε

P = 1.

21

ESTIMATES ON SUBHARMONIC FUNCTIONS

and ∂x v = H[ν] ∂x (v − Pε v) = H[ν − Pε ν] Thus v − Pε vBMO ≤ ∂x−1 [ν − Pε ν]∞ = max | χJ − χJ ∗ Pε , ν| J


ε0 ] < ε1

Then √ √ −c[ ε0 + mes[x ∈ T |u(x) − u| > ε0 ] < e

Proof. From (4.9) and the John-Nirenberg inequality.

ε1 B −1 ε0 ]

(4.11)

22

CHAPTER 4

We also will need estimates on functions of several variables with pluri-subharmonic extension. We only treat the case of two variables in Lemma 4.12 below. Similar inequalities for d variables are obtained along the same lines. Thus, in the remainder of this chapter, assume that u = u(x, y) is 1-periodic in x, y, |u| ≤ 1, and with pluri-subharmonic extension, u = u ( x, y ), |Im x| ≤ 1, |Im y| ≤ 1 satisfying | u| < B Lemma 4.12. Let u be as above, satisfying, moreover, mes[(x, y) ∈ T2 |u(x, y) − u| > ε0 ] < ε1

Then







2

mes[(x, y) ∈ T |u − u| >

1/4 ε0 ]

−c

< exp

 1/4 ε0

+

B ε0

−1 

1/2 1/4 ε1

(4.13) Proof. We may assume that u = 0. Write u = u0 +u1 , u0 ∞ < ε0 , u1 1 < ε1 . Denote  √ A = {y ∈ T |u1 (x, y)|dx < ε1 } Hence

√

mes (T\A) < Denote

ε1

 u(x, y)dx

U (y) = T

Thus, for y ∈ A, we have |U (y)| < ε0 +

√ ε1

and from the one-variable result applied in x, we have 1/4

u(·, y)BMO ≤ C(ε0 + ε1 B 1/2 ) and hence



c

e Integrating in y ∈ A,

 

|u(x,y)| 1/4 1/2 ε0 +ε B 1

c

e

|u(x,y)| 1/4 1/2 ε0 +ε1 B

dx < C

dxdy < C

A

Denote γ=e

−c

1/2 ε0 1/4 1/2 ε0 +ε1 B

23

ESTIMATES ON SUBHARMONIC FUNCTIONS



B = {x ∈ T

e

|u(x,y)| c 1/4 1/2 ε0 +ε1 B

dy < γ −1 }

A

Thus mes (T\B) < Cγ For x ∈ B, write

1/2 mes [y ∈ T |u(x, y)| > 2ε0 ] < mes A + γ c

√ ε1 + e −c

−1

e

−2c

1/2 ε0 1/4 1/2 ε0 +ε1 B


ε0 ] < e Therefore, 1/4

mes [|u| > ε0 ] < mes (T\B) + e−c[ε0 1/4

N −1/10 , 1 2 mes[x ∈ T log MN (x) − LN (E) > κ] < Ce−cκ N N

(5.3)

Remark. If (5.2) is weakened to a DC kω > c|k|−A for k ∈ Z\{0} we still get a conclusion 1 σ mes [x ∈ T log MN (x) − LN (E) > N −σ ] < Ce−N N

(5.4)

for some σ = σ(A) > 0. Proof of Theorem 5.1. By assumption, MN (x) has an analytic extension MN (z) to a strip |Im z| < ρ, for some ρ > 0, satisfying MN (z) < C N . Hence 1 log MN (x) N has bounded subharmonic extension to |Im z| < ρ, and therefore, u(x) =

< |ˆ u(k)| ∼

1 |k|

It is also clear from the definition of u and MN (x) that |u(x) − u(x + ω)|
Write

1 if 1 ≤ j < q 2q

   R − |j| R (R ∼ κN ) u(x + jω) + 0 R2 N |j|

1 log λ 2

Thus, compared with Proposition 3.3, Proposition 7.2 holds in any dimension, but the proof below requires a DC on ω. See also the comments at the end of this chapter. We will use the following consequence of the classical Lojasiewicz result. Lemma 7.3. There is a constant c0 = c0 (v0 ) s.t. mes[x ∈ Td |v0 (x) − E1 | < δ] < δ c0

for all E1 and suffi ciently small δ > 0. Proof of Proposition 7.2. Fix a large scale n0 and chose − c2

δ = n0

0

and λ = δ −10

(7.4)

32

CHAPTER 7

Choice of n0 depends on diophantine assumptions on ω. If v = λv0 , it follows from (7.4) that for n ≤ n0 and E arbitrary, Mn (E, x) > (δλ − 1)n for x outside a set of measure at most nδ c0 < n10 . Hence,    1 1 log Mn (E, x)dx > 1 − log(δλ − 1) log(λ + |E|) + C > Ln (E) = n n0 4 > log λ 5 Fix E, |E| < Cλ (otherwise there is nothing to prove). Using the submultiplicity L2n (E) ≤ Ln (E), the preceding permits us to find n1 < n0 , n1 ∼ n0 satisfying |L2n1 (E) − Ln1 (E)|
κ log λ] < e−cN N Take κ = 10−2 . If σ

c

n2 = mn1 ∼ e 2 n1

the LDT permits us to ensure that 1 1 Ln (E) max log Mn1 (x + jω) − Ln1 (E) < 1≤j≤n2 n1 50 1 and

1 1 log M2n1 (x + jω) − L2n1 (E) < max L2n1 (E) 1≤j≤n2 2n1 50

except for x ∈ Ω ⊂ Td , mes Ω < e− 2 n1 c

σ

Fix x ∈ Ω and define for k = 1, · · · , m Ak = Mn1 (x + (k − 1)n1 ω) ∈ SL2 (R) We verify the conditions of Proposition 6.1. By (7.6) 49

3

Ak  > en1 50 Ln1 (E) = μ > e 4 (log λ)n1 > m

(7.6)

(7.7)

33

CONSEQUENCES FOR LYAPOUNOV EXPONENT, IDS, AND GREEN’S FUNCTION

and by (7.6), (7.7), and (7.5) log Ak  + log Ak+1  − log Ak+1 Ak  < |2n1 Ln1 (E) − 2n1 L2n1 (E)| + < 15 n1 Ln1 (E)
δ0 = fi xed constant. Then for all n ∈ Z+ large enough |L(E) + Ln (E) − 2L2n (E)| < e−cn

σ

(7.13)

where σ is the exponent in (7.1) and c = c(δ0 ) > 0. Proof. The preceding argument shows that if we take n = n1 large enough (depending on δ0 ) and log n2 ∼ nσ , then |Ln2 (E) + Ln (E) − 2L2n (E)| < e−cn

σ



 see (7.9) and also

|L(E) − Ln2 (E)| ≤



|Lns+1 (E) − Lns (E)|

s≥2

≤2



|L2ns (E) − Lns (E)| + e−cns

σ



s≥2

δ0 for all E ∈ [E1 , E2 ]. Then, for E, E  ∈ [E1 , E2 ], |L(E) − L(E  )| < C exp[−c(log |E − E  |−1 )σ ]

(7.15)

In particular, if d = 1 (and ω typical), |L(E) − L(E  )| < C|E − E  |κ

(7.16)

for some κ = κ(δ0 ) > 0. Proof. Obviously, |Ln (E) − Ln (E  )| < C n |E − E  | so that by (7.13) |L(E) − L(E  )| < C n |E − E  | + e−cn

σ

and (7.15) follows from choice n ∼ log

1 |E−E  | .

Corollary 7.17. Assume that ω satisfi es a DC. Then L(E) is a continuous function of E ∈ R. Proof. We claim that if L(E) > δ, then also L(E  ) > 2δ for |E − E  | < ε(δ). This will in particular imply Corollary 7.17. Choose sufficiently large n1 (depending in particular on δ) to ensure that |L2n1 (E) − Ln1 (E)| < 10−3 Ln1 (E) For |E  − E| < C −n1 δ = ε(δ), also Ln1 (E  ) >

9 δ and |L2n1 (E  ) − Ln1 (E  )| < 10−2 Ln1 (E  ) 10

CONSEQUENCES FOR LYAPOUNOV EXPONENT, IDS, AND GREEN’S FUNCTION

35

Following again the proof of Proposition 7.2, we get |L(E  ) − Ln1 (E  )| < 10−1 Ln1 (E  ) + e−cn1

σ

and hence L(E  ) >

3 δ Ln1 (E  ) > 4 2

proving the claim. Relating the Lyapounov exponent L(E) and the integrated density of states (IDS) N (E), the following regularity property is obtained for shift models with typical ω. Proposition 7.18. Assume L(E) > δ0 for E ∈ [E1 , E2 ]. For d = 1, the IDS is Holder continuous on [E1 , E2 ]. Thus, for some κ = κ(δ0 ) > 0, |N (E) − N (E  )| < C|E − E  |κ for E, E  ∈ [E1 , E2 ]

For d ≥ 2, we have

 |N (E) − N (E )| < C exp[−c log 

1 |E − E  |

σ ]

for E, E  ∈ [E1 , E2 ]

Proof. Recalling the Thouless formula  L(E) = log |E − E  |dN (E  ) we see that L(E) and N (E) are related by the Hilbert transform. Thus the claim follows from Corollary 7.14. Remarks. 1. Consider the Almost Mathieu operator H = λ cos(x + nω)δnn + Δ in perturbative regime (i.e., λ large). It may then be shown that for typical ω, we have |N (E) − N (E  )| < Cκ |E − E  |κ for all κ < 12 . The exponent κ = 12 is optimal because of the presence of gaps in the spectrum. 2. Concerning Proposition 7.2, the author established more recently the full analogue of Proposition 3.3, thus the Sorets-Spencer theorem with λ > λ0 (v) for d > 1 (see [Bo] in Chapter 3). Problem. Does Proposition 7.18 require the positivity assumption of the L(E)? What happens in the (nonperturbative) Almost Mathieu model, say, at λ = 2? A (negative) result on this issue will be pointed out in the next chapter. Returning to Chapter 2, we state the following important consequence of the LDT to Green’s function estimates (in the shift model). Recall (2.7) |GN (E, x)(n1 , n2 )| < and the subsequent discussion.

Mn1 (x, E) · MN −n2 (T n2 x, E) | det[HN (x) − E]|

36

CHAPTER 7

Proposition 7.19. Assume L(E) > δ0 . Then for N > N0 (δ0 ), there is a set Ω ⊂ Td satisfying for some σ > 0 mes Ω < e−cN

σ

(7.20)

and such that for any x outside Ω, one of the intervals Λ = [1, N ]; [1, N − 1]; [2, N ]; [2, N − 1]

will satisfy |GΛ (E, x)(n1 , n2 )| < e−L(E)|n1 −n2 |+N

1−

(7.21)

CONSEQUENCES FOR LYAPOUNOV EXPONENT, IDS, AND GREEN’S FUNCTION

37

References [B] J. Bourgain. Positivity and continuity of the Lyapounov exponent for shifts on Td with arbitrary frequency vector and real analytic potential, preprint 2002. [B-J] J. Bourgain, S. Jitomirskaya. Continuity of the Lyapounov exponent for quasi-periodic operators with analytic potential, J. Stat. Phys. 108(5–6) (2002), 1203–1218.

Chapter Eight Refinements The purpose of this chapter is to analyze in more detail the estimates of Chapter 7 for the small Lyapounov exponent L(E). We consider only the case of the 1-frequency shift model H(x) = v(x + nω )δnn + Δ

(x, ω ∈ T)

(8.1)

with v 1-periodic and with bounded analytic extension on z = x + iy, |y| ≤ 1, say. Assume again the rotation number ω satisfying (5.2) 1 kω > c for k ∈ Z\{0} (8.2) |k|[log(1 + |k|)]3 (this assumption may be replaced by weaker ones). Proposition 8.3. Assume that the Lyapounov exponent L(·) of (8.1) satisfi es L(E) > 0 for E ∈ [E1 , E2 ] ⊂ R

Then L(·) and the IDS N (·) are Holder continuous on [E1 , E2] with exponent c > 0 depending only on the bound for the analytic extension of v assuming ω satisfi es  (8.2) . Without making any positivity assumption on L(·), we may state Proposition 8.4. For any A > 0, we have the unconditional estimate −A  1  |N (E) − N (E )| ≤ CA log (8.5) |E − E  | for |E − E  | < 12 . Proposition 8.4 improves on the general log-Hölder regularity result for stochastic Jacobi matrices (see [C-S]). There is the following immediate corollary of Proposition 8.3 for the Almost Mathieu operator. Corollary 8.6. For λ > 2, the IDS of H(x) = λ cos(x + nω)δnn + Δ

(8.7)

is Hölder continuous with exponent c > 0 independent of λ. Remark. The constant K in the inequality |N (E) − N (E  )| < K|E − E  |c does depend on the lower bound on L(E), however. Otherwise, the IDS of (8.7) at λ = 2 would be Hölder continuous. Now let 1 ω= a1 + a2 + 1 1 a3 + .. .

40

CHAPTER 8

be the continuous fraction expansion of ω, and assume that all convergents satisfy |as | > C(ε)

(8.8)

where C(ε) is a sufficiently large constant depending on given ε > 0. According to the results from Helffer-Sjostrand [H-S], one may then write  Ij with a1 ∼ a1 Spec Hλ=2 ⊂ I0 ∪ 1≤j≤a1 1

where {Ij } are intervals, |I0 | < ε and |Ij | < e− 2 a1 for j ≥ 1. Furthermore, there is a renormalization of H on each Ij , j ≥ 1 with a similar description of the spectrum, a2 replacing a1 , etc. If ε > 0 is small enough, Var N (I0 ) < 12 , and hence Var N (Ij ) > 2a1 for some 1 j ≥ 1. Iterating, one obtains after s steps an interval I satisfying 1

|I| < e− 2 (a1 +···+as ) while Var N (I) > 2−s

a1

1 · · · as

If thus 1s (a1 + · · · + as ) → ∞, which is typically the case, N clearly cannot be Hölder regular. We will first prove Proposition 8.3, and a more careful examination of the arguments will then lead to Proposition 8.4. In the next few lemmas we establish upper bounds on the transfer matrix MN (x, E), valid for all x ∈ T. Lemma 8.9. One has for all x ∈ T the upper bound  R−1  1  1 (log R)5 log MN (x + jω; E) < LN (E) + c R j=0 N R

(8.10)

Proof. Denoting 1 log MN (x, E) N recall the Riesz-decomposition (4.2)  u(x) = log |x − w|μ(dw) + h(x) for x ∈ [−1, 1] u(x) =

with μ ≥ 0 and h smooth. Fix δ > 0 and majorize u by the periodic function  vδ (x + j)η(x + j) (8.11) u(δ) (x) = j∈Z

with v(δ) on [−1, 1] defined by  v(δ) (x) = log(|x − w| + δ)μ(dw) + h(x)

41

REFINEMENTS

and 0 ≤ η ≤ 1 a smooth function satisfying supp η ⊂ ] − 1, 1[ and



η(x + j) = 1

Thus clearly u ≤ u(δ) pointwise Since for k ∈ Z

  u(δ) (k) =

R

v(δ) (x)η(x)e−2πikx dx ≡ v δ η(k)

we obtain  u(δ) (0) = u η(0) + 0((u − vδ )η1 ) =u (0) + 0(δ. log 1δ ) and for k = 0

 |ˆ u(δ) (k)| < C min

1 1 , |k| k 2 δ

(8.12)

 (8.13)

The left side of (8.10) is bounded by 1 R

R−1 

u(δ) (x + jω) =

j=0

u ˆδ (0) +





1 R

u ˆδ (k)

k=0



LN (E) + 0 δ. log

1 δ

R−1 

 e

2πikjω

  e2πikx = by (8.12)

j=0

+

 k=0

 1 | uδ (k)| 1+Rkω

(8.14)

Let {qs }s≥1 be the approximants of ω satisfying by (8.2) qs < qs−1 (log qs )3

(8.15)

Fixing some s∗ , estimate using (8.13)  1 | uδ (k)| 1+Rkω ≤ k=0

c





s≤s∗ qs−1 ≤|k| 0 independent of the smallness of the Lyapounov exponent. Together with the Thouless formula, this proves Proposition 8.3. The preceding does not answer the question of which regularity properties N (E) has without the assumption that L(E) > 0. The method used above permits us to show Proposition 8.4. We give a sketch of the argument. Fix an energy E s.t. L(E) > 0. Let γ > 0 be a small constant. Take N 0 ∈ Z+ satisfying   (8.37) N01−γ LN0 (E) ≥ max M 1−γ LM (E), 1 M L N0 (E) 2 2 and hence 21−γ LN0 (E) > L N0 (E) ≥ LN0 (E) 2

(8.48)

γ/6

Thus, for x outside a set of measure < e−N0 ,   N0 L N0 1 − o(1) = log μ log M N0 (x; E) > 2 2 2

(8.49)

46

CHAPTER 8

and

log M N0 (x) + log M N0 (x + N0 ω) − log MN0 (x) < 2 2 2  N0 L N0 (E) − LN0 (E) + o(1)N0 L N0 (E) < 2    2  N0 L N0 (E) 1 − 2γ−1 + o(1) < 1 − γ2 log μ

(8.50)

2

Estimate (8.50) is weaker than condition (6.3) in the assumptions from Proposition 6.1. One verifies that the argument may still be carried out, provided n < μ γ/2 and with bound nμ−γ/2 in (6.4). This thus permits us to show that N0 |LN (E) + L N0 (E) − 2LN0 (E)| < c LN0 (E) (8.51) 2 N 1

γ/6

for N < e 2 N0 , N0 |N . In particular, LN (E) > 2LN0 (E) − L N0 (E) − o(γ)LN0 (E) > γLN0 (E)

(8.52)

2

γ/6 1 2 N0

if γ −2 N0 < N < e , N0 |N . γ/10 Continuing the argument with N0 replaced by N1 < eN0 etc. shows that γ/10

|L(E) + L N0 (E) − 2LN0 (E)| < e−N0

(8.53)

2

L(E) > γLN0 (E) Also, if N0 |N and LN (E) − L2N (E)
L(E)− 1−γ x

1 − 1−γ

max(log MN (x, E)) < 4γ −1 (N + L(E) x

)L(E) for all N

(8.56)

Next, take E  such that τ ≡ |E − E  | < e−C(γ)L(E)

1−

1 1−γ

with C(γ) sufficiently large. One may then find N > conditions

(8.57) 1

1

L(E) 1−γ

satisfying the

N0 = N0 (E)|N e−C(γ)N L(E) < |E − E  | < e LN (E) − L2N (E)
0 log N

(8.65)

We may then find (arbitrarily large) N0 ∈ Z+ satisfying (8.37) for some γ > 0. From (8.54), L(E) > 0. Thus (8.65) implies positivity of the Lyapounov exponent.

48

CHAPTER 8

References [C-S] W. Craig, B. Simon. Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices, CMP 90(2) (1983), 207–218. [H-S] B. Helffer, J. Sjostrand. Analyse semi-classique pour l’equation de Harper, Memoire SMF 34 (1988).

Chapter Nine Some Facts about Semialgebraic Sets The purpose of this chapter is to summarize a number of results from the literature for later use. Some slightly weaker statements (which do suffice for our needs) also may be found in Bourgain and Goldstein [5] (in Related References) with proof. Definition 9.1. A set S ⊂ Rn is called semialgebraic if it is a finite union of sets defined by a finite number of polynomial equalities and inequalities. More precisely, let P = {P1 , . . . , Ps } ⊂ R[X1 , . . . , Xn ] be a family of real polynomials whose degrees are bounded by d. A (closed) semialgebraic set S is given by an expression   S= {Rn |P sj 0}, j ∈Lj

where Lj ⊂ {1, . . . , s} and sj ∈ {≥, ≤, =} are arbitrary. We say that S has degree at most sd, and its degree is the infimum of sd over all representations as in (9.1). The projection of a semialgebraic set of Rk+ onto Rk is semialgebraic. This is known as the Tarski-Seidenberg principle; see Bochnak, Coste, and Roy [4]. The currently best quantitative version of this principle is due to Basu, Pollack, and Roy [3] and Basu [1]. For the history of such effective Tarski-Seidenberg results, we refer the reader to those papers. Proposition 9.2. Let S ⊂ Rn be semialgebraic defi ned in terms of s polynomials of degree at most d as in (9.1). Then there exists a semialgebraic description of its projection onto Rn−1 by a formula involving at most s2n dO(n) polynomials of degree at most dO(n) . In particular, if S has degree B , then any projection of S has degree at most B C , C = C(n). This is a special case of the main theorem in Basu, Pollack, and Roy [3]. Another fundamental result on semialgebraic sets is the following bound on the sum of the Betti numbers by Milnor, Oleinik, and Petrovsky, and Thom. Strictly speaking, their result applies only to basic semialgebraic sets, which are given purely by intersections without unions. The general case as in Definition 9.1 above was settled by Basu [2]. Theorem 9.3. Let S ⊂ Rn be as in (9.1). Then the sum of all Betti numbers of S n is bounded by sn O(d) . In particular, the number of connected components of  n S does not exceed sn O(d) . This is a special case of Theorem 1 in Basu [2]. Another result that we shall need is the following triangulation theorem of Yomdin [7], later refined by Gromov [6]. We basically reproduce the statement of that result from Gromov [6; see p. 239]. Theorem 9.4. For any positive integers r, n there exists a constant C = C(n, r) with the following property: Any semialgebraic set S ⊂ [0, 1]n ⊂ Rn can be

50

CHAPTER 9

< (deg S + 1)C simplices, where for every closed k -simplex triangulated into N ∼ Δ ⊂ S there exists a homeomorphism hΔ of the regular simplex Δk ⊂ Rk with unit edge length onto Δ such that hΔ is real analytic in the interior of each face of Δ. Furthermore, Dr hΔ  ≤ 1 for all Δ.

Related References [1] S. Basu. New results on quantifier elimination over real closed fields and applications to constraint databases, J. ACM 46(4) (1999), 537–555. [2] S. Basu. On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets, Discrete Comput. Geom. 22(1) (1999), 1–18. [3] S. Basu, R. Pollack, M.-F. Roy. On the combinatorial and algebraic complexity of quantifier elimination, J. ACM 43(6) (1996), 1002–1045. [4] J. Bochnak, M. Coste, M.-F. Roy. Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 36, Springer-Verlag, Berlin, 1998. [5] J. Bourgain, M. Goldstein. On non-perturbative localization with quasi-periodic potential, Annals of Math. (2) 152(3) (2000), 835–879. [6] M. Gromov. Entropy, homology and semialgebraic geometry, Séminaire Bourbaki, 1985/86(663), Astérisque 145–146 (1987), 5, 225–240. [7] Y. Yomdin. C k -resolution of semi-algebraic mappings. Israel J. Math., 57(3) (1987), 301–317. In our applications, the number n of variables always will be bounded. Returning to Theorem 9.4, we notice the following corollaries. Corollary 9.5. (i) Let S ⊂ [0, 1]n be connected and semialgebraic of degree B . If p, q ∈ S , there is a path γ : [0, 1] → S s.t. γ(0) = p, γ(1) = q and |γ| ˙ < BC .  (ii) Let S ⊂ nj=1 [0, ρj ]. Then, for p, q ∈ S , there is a path γ : [0, 1] → S s.t. γ(0) = p, γ(1) = q and n  |γ˙ j |

ρj j=1   (ii) follows from simple rescaling .

< BC

Corollary 9.6. Let S ⊂ [0, 1]n be semialgebraic of degree B . Let ε > 0 be a small number and mesn S < εn . Then S may be covered by at most B C ( 1ε )n−1 ε-balls. Proof. From the assumption, clearly, dist(p, ∂S) < ε for all p ∈ S. By Theorem 9.4, ∂S is obtained as an image of at most B C sufficiently smooth maps defined on regular simplices Δk with k ≤ n − 1. The conclusion is clear.

51

SOME FACTS ABOUT SEMIALGEBRAIC SETS

The next fact deals with the intersection of a semialgebraic set of small measure and the orbit of a diophantine shift. Corollary 9.7. Let S ⊂ [0, 1]n be semialgebraic of degree B and mesn S < η . Let ω ∈ Tn satisfy a DC and N be a large integer, 1 log B log N < log η Then, for any x0 ∈ Tn #{k = 1, · · · , N |x0 + kω ∈ S(mod 1)} < N 1−δ

(9.8)

for some δ = δ(ω). Proof. Choose ε > η 1/n + N −δ . Then S is covered by at most B C ( 1ε )n−1 εballs, and the orbit occupation of a single ball is at most Cεn N (since N is large enough). This follows from the fact that ω was assumed diophantine and standard equidistribution considerations. The statement may be seen as follows. Let χ be the indicator function of the ball 1 B(0, ε) centered at 0. If R = 10ε and FR is the usual (1-dim) Fejer-kernel, one has n that χ ≤ CR−n j=1 FR (xj ). Therefore, passing to Fourier transform, N N    ik.ω −n   χ(xo + kω) ≤ CR e FR (1 ) · · · FR (n ) 1

0≤|1 |,...,|n | c|k|−A for k ∈ Zd \{0}

(10.2)

Assume that the Lyapounov exponent L(E) = Lω (E) > c0

(10.3)

for all ω ∈ DC and E ∈ R. Fix x0 ∈ Td . Then, for almost all ω ∈ DC , Hω (x0 ) satisfi es Anderson localization (i.e., H has p.p. spectrum with exponentially localized states). Remarks. 1. The case d = 1, 2 was treated in [B-G] (see Chapter 1). The argument presented below makes more extensive use of semialgebraic set theory to deal with certain transversality issues and applies to arbitrary d. 2. This result is nonperturbative. Assume, for instance, that v0 is a nonconstant trigonometric polynomial on Td , and let v = λv0 . Herman’s lower bound permits us then to specify λ ≥ λ0 (v) for (10.3) to hold, independently of ω. 3. In Theorem 10.1, the frequency vector ω also is considered as a parameter. The set of “good ω’s” in the conclusion of Theorem 10.1 requires, at least in our argument, a further exclusion of a zero-measure set, for which we do not have a simple arithmetic description. In some cases, for instance, in the Almost Mathieu case, v(x) = λ cos x, an explicit arithmetic condition may be stated. We will discuss this at the end of this chapter. There are the following two main ingredients in the proof of Theorem 10.1: 1. The LDT for the fundamental matrix and Proposition 7.19 on the Green’s function estimate 2. Semialgebraic set theory as described in Chapter 9 As was already mentioned, to establish AL for H, it suffices to show that any extended state is exponentially decaying. Thus, if ξ = (ξn )n∈Z and E ∈ R satisfy |ξn | < C|n| for |n| → ∞ and Hξ = Eξ

56

CHAPTER 10

then |ξn | < e−c|n| for |n| → ∞

(10.4)

(In fact, it can be shown that the exponent c in (10.4) may be taken L(E) − ε for all ε > 0.) Fix N0 and consider the property 1−

|GN0 (E, x)(n1 , n2 )| < e−c0 |n1 −n2 |+N0 for all 1 ≤ n1 , n2 ≤ N0 (10.5)  ik.x −ρ|k| Writing v(x) = ˆ(k)e , |ˆ v (k)| < e , it is clear that in (10.5) we k∈Zd v  basically may substitute v by v1 (x) = |k|

N0 2

N0

by (10.10), and (10.12)

many.

I = [−j0 + 1, j0 − 1] and write  RI H(x0 ) − E)RI ξ = −(ξ−j0 e−j0 +1 + ξj0 ej0 −1 ) 1 = |ξ0 | ≤ |GI (x0 , E)(0, j0 − 1)| |ξj0 | + |GI (x0 , E)(0, −j0 + 1)| |ξ−j0 | ≤ GI (x0 , E)(|ξj0 | + |ξ−j0 |) If j0 , −j0 satisfy both (10.12), we conclude that G]−j0 ,j0 [ (x0 , E) > e

c0 2

N0

(10.13)

or equivalently   c0 dist E, SpecH]−j0 ,j0 [ (x0 ) < e− 2 N0

(10.14)

Thus, if there is an extended state ξ, ξ0 = 1 with energy E, then, for any large N0 , there is some j0 , |j0 | < N1 = N0C for which (10.14) holds. Denote  E = Eω = Spec H]−j,j[ (x0 ) (10.15) |j|≤N1

It clearly follows from (10.11) and (10.14) that if  Ω(E  ) x ∈

(10.16)

E  ∈Eω

then one of the sets Λ in (10.10) satisfies 1−

|GΛ (E, x)(n1 , n2 )| < e−c0 |n1 −n2 |+N0 for n1 , n2 ∈ Λ Next, let N2 = N0C



(10.17)

58

CHAPTER 10

with C  a sufficiently large constant, and suppose that we ensured that  1/2 x0 + nω ∈ Ω(E  ) (mod 1) for all N2 < |n| < N2

(10.18)

E  ∈Eω 1/2

Thus, for each N2

< |n| < 2N2 , there is an interval

Λ(n) ∈ {[−N0 , N0 ], [−N0 , N0 − 1], [−N0 + 1, N0 ], [−N0 + 1, N0 − 1]} for which (10.17) holds: 1−

|GΛ(n) +n (E, x0 )(n1 , n2 )| < e−c0 |n1 −n2 |+N0 for n1 , n2 ∈ Λ(n) + n Define the interval



= Λ 1/2

N2

(10.19)

1/2

(Λ(n) + n) ⊃ [N2 , 2N2 ]

N 2 , max |ProjL ej | < ε2

0≤j 0. In the second inequality, use the fact that  1 | cos x − E0 | 2 − < C T

Next, a minoration on (10.16) is needed. There is always the obvious lower bound  ˆ 1 )] > s log ε0 log[| cos(x + kω) − E0 | + 2ε0 (1 + φ k∈γ

where s = |γ| 1 10

Assume s > ε0 N . Then a better lower bound will be given. Since ω in DC, we have for N large enough, κ > log1N , that #{k = 0, 1, . . . , N − 1 kω + x ± θ0  < κ} < 10κN Letting κ ∼

s N,

and thus

elements k ∈ γ  s 2 1/4 > log ε0 log[· · · ] > log κ2 > log 10−3 N

it follows that for at least



log[· · · ] >

k∈γ

s 2

s s 3 1/4 log ε0 + log ε0 > s log ε0 2 2 4

In summary, 1

−

log(11.14) < −(log 2)N + ε02 N + s. log

1 ε0

(11.17)

1

and if s > ε010 N , 1 1 3 − log(11.14) < −(log 2)N + ε02 N + s log 4 ε0 Returning to (11.13), we obtain from the preceding |Ann | <    s 1−   s b 2 −N εs−1 e−ρb eε0 N ε10 2 2 s−1 b≥|n−n | s≤b

+2

−N



1 s≤ε 10 0

b≥|n−n |



s≤b 1 s>ε010 N

   34 s 1− b 2 εs−1 e−ρb eε0 N ε10 s−1

 2

s

(11.18)

70

CHAPTER 11 1 20

1 20

and distinguishing the cases b ≤ ε0 N, b ≥ ε0 N , for ε0 = ε0 (ρ) small enough 1− 2 N

< 2−N eε0

1 20

(2ε0

N



ρ



.e−ρ|n−n | + e− 2 |n−n | )

(11.19) √ From (11.11) and (11.19), it results that for x ∈ ΩN (E), there is some m, |m| < N for which |GN (E + i0, x + mω)| < eN

1−σ

1

 +ε020 N − ρ 2 |n−n |

Thus (11.5) holds. This proves Proposition 11.4. Theorem 11.2 generalizes to Theorem 11.20. Let φ and v be real analytic on T, v nonconstant. Then, for 0 < ε < ε0 , ε0 = ε0 (v, φ), Hω (x) = v(x + nω)δnn + εSφ

satisfi es A.L. for (x, ω) ∈ T2 in a set of full measure. Again, the issue is Proposition 11.4, under the assumptions of Theorem 11.20. Analyzing the proof of Proposition 11.4, the main problem is to obtain the “correct” lower bound on N1 T log | det[HN (x) − E]|dx. In the special case v(x) = cos x, this is achieved by Herman’s argument. In other situations (with v(x) given by a trigonometric polynomial), that argument would not provide the right estimate. Using a variant on the proof of Proposition 3.3, we show the following: Proposition 11.21. Under the assumptions of Theorem 11.20, we have for diophantine ω   1 log | det[HN (x) − E]|dx > log |v(x) − E|dx − κ(ε) (11.22) N T T where κ(ε) → 0 for ε → 0 and taking N large enough. Proof. Assume |ˆ v (k)| < e−|k|ρ (k ∈ Z), ρ > 0. Since v is nonconstant,  1 log |v(x) − E|dx > −Cv (11.23) min E

0

for some constant Cv . Fix 0 < σ < ρ2 . Proceeding as in the proof of Proposition 3.3, there is δ0 > 0 s.t. min |v(x ± iy0 ) − E| > δ0

min σ max E

2

1 + log |v(x + iy0 ) − E|dx − C + y1 − y0 δ0 ρ 0 0  1 σ 1/2 > log |v(x + iy0 ) − E|dx − Cv − Cε0 ρ 0 Subject to replacement of y0 by −y0 , (see (11.24)), and since    1 1 1 log |v(x + iy0 ) − E|dx + log |v(x − iy0 ) − E|dx log |v(x) − E|dx ≤ 2 2 0 it follows that     1 1 σ 1/2 + ε0 log | det[HN (x) − E]|dx > log |v(x) − E|dx − C N 0 ρ This proves Proposition 11.21. In order to get an upper bound on  | det[Rγc HN (x) − E)Rγc ]| where γ ⊂ {0, 1, . . . , N − 1}, we use the following. Lemma 11.29. If γ ⊂ {0, 1, . . . , N − 1} and |γ| = s > ε1 (ε0 )N where log ε10 ∼ log ε11 , then for all x  3 log[|v(x + kω) − E| + ε0 ] > s log ε0 (11.30) 4 k∈γ

73

GENERALIZATION TO CERTAIN LONG-RANGE MODELS

Proof. Approximation by trigonometric polynomials permits for fixed κ to get  ikx w(x) = |k| 0. Thus for N large enough #{k = 0, 1, . . . , N − 1 |v(x + kω) − E| < κ} < 2κc N and appropriate choice of κ implies  s s s log[|v(x + kω) − E| + ε0 ] > log ε0 + C log 2 2 N k∈γ

The lemma follows. Remarks. 1. Proposition 11.21 implies the following refinement of Proposition 3.3 (assuming ω diophantine). Proposition 11.31. Let H(x) = λv(x + nω)δnn + Δ (1-frequency on Z) with v nonconstant real analytic and ω diophantine. Then, for λ > λ0 (v), we have  L(E) > log |λv(x) − E|dx − κ(λ) T

where κ(λ) → 0 for λ → ∞. 2. The method described above permits us also to establish localization results for band-Schrödinger operators with index set Z × {1, . . . , b}

(11.32)

Thus, let H(ω, θ) be the following lattice Schrödinger operator on (11.32): H(n,s),(n ,s ) (ω, θ) = λvs (θ + nω)δnn δss + Δ

(11.33)

where {vs |s = 1, . . . , b} are real analytic, nonconstant on T, and Δ stands for the Laplacian on Z2 $ 1 if |n − n | + |s − s | = 1   Δ((n, s), (n , s )) = = 0 otherwise Theorem 11.34. Consider H as above. Then for λ > λ0 (v1 , . . . , vb ), Anderson localization holds for (ω, θ) ∈ T2 in a set of full measure. See [B-J1]. Remark. In the case of quasi-periodic Schrödinger operators on Z2 H(ω1 , ω2 , ; θ1 , θ2 ) = λv(θ1 + n1 ω1 , θ2 + n2 ω2 ) + Δ (v real analytic on T2 ), only perturbative localization results may be expected (see [BGS] and Chapter 17).

74

CHAPTER 11

References [B-J1] J. Bourgain, S. Jitomirskaya. Anderson localization for the band model, Springer LNM 1745 (2000), 67–79. [B-J2] Idem. Non-perturbative absolutely continuous spectrum for 1D quasiperiodic operators, Inventiones Math. 148(3) (2002), 453–463. [BGS] J. Bourgain, M. Goldstein, W. Schlag. Anderson localization on Z 2 for Schrödinger operators with quasi-periodic potential, Acta Math. 188 (2002), 41–86.

Chapter Twelve Lyapounov Exponent and Spectrum First, we recall some basic facts from spectral theory. Let H be a bounded self-adjoint operator on 2 (Z). Then, for z ∈ C\Spec H, (H − z)−1 is analytic (hence in particular for Im z > 0), and we have for f ∈  2 Im (H − z)−1 f, f  = Im z.(H − z)−1 f 2

(12.1)

Thus φf (z) = (H − z)−1 f, f  is an analytic function on the upper half plane with Im φf ≥ 0 (φf is a so-called Herglotz function). Therefore, one has a representation  1 φf (z) = (H − z)−1 f, f  = μf (dλ) λ − z R where μf is the spectral measure associated to f . Thus μf ≥ 0, μf  = f 2 . If for f, g ∈ 2 we let 1 μf,g = [μf +g − μf −g + i(μf +ig − μf −ig )] 4 then  1

(H − z)−1 f, g = μf,g (dλ) λ − z R Decompose μf = μf,pp + μf,sc + μf,ac is discrete,singular continuous, and absolutely continuous parts, Define pp = pp (H) = complement open set R for which   maximal G in   μf,pp (G) = 0, ∀f and similarly sc , ac . Let = ∪ sc ∪ ac = pp Spec H = {E ∈ R inf f =1 (H − E)f  = 0}. Next, we will recall some facts from Kotani’s theory, such as the Ishii-Pastur-Kotani theorem. We consider the context of Schrödinger Hamiltonians on Z associated with a shift H(x) = v(x + nω)δnn + Δ

(12.2)

(ω DC), as studied earlier. The results mentioned below apply in much greater generality, however. There are also several other important facts from this theory that will not be brought up here. Denoting again L(E) the Lyapounov exponent, recall that in the context (12.2), L(E) is a continuous function of E, and in particular, [E ∈ R L(E) = 0] is closed.

76

CHAPTER 12

Proposition 12.3. (Pastur-Ishii) x x [L(E) > 0] ∩ = φ hence ⊂ [L(E) = 0] ac ac    x almost surely. (Notice that pp , ac , sc do depend on x.) Proof of Proposition 12.3. Consider the set Ex = {E ∈ R H(x) has a generalized eigenstate ξ with eigenvalue E} This means that



 H(x) − E ξ = 0

(12.4)

where, say, |ξn | < |n| for |n| → ∞ It is well known that for f ∈ 2 , μxf (Ex ) = 1 On the other hand, if L(E) > 0, then the Green’s function G[−N,N ] (E + io, x) satisfies an estimate  N |G[−N,N ] (E + io, x)(n, n )| < e−c|n−n | for |n − n | > 10 except for x ∈ ΩN (E), mes ΩN (E) < N12 , for N large. Assume (12.4). Fix j0 s.t. ξj0 = 0, and let N > 2j0 . If x ∈ ΩN (E), we get 0 < |ξj0 | ≤ |GN (E, x)(j0 , N )| |ξN +1 | + |GN (E, x)(j0 , −N )| |ξ−N −1 | < e−cN (|ξN +1 | + |ξ−N −1 |) In particular, for L(E) > 0, x ∈ lim ΩN (E) ⇒ E ∈ Ex and thus Write  T

mes [(x, E) ∈ T × R L(E) > 0 and E ∈ Ex ] = 0

μxf,ac [E L(E) > 0]dx = sup

K>0

x   dμf,ac 0, dE T



where, by the preceding, the integral is  mes [E L(E) > 0, E ∈ Ex ]dx = 0 ≤K T

Hence, x almost surely

μxf,ac [E L(E) > 0] = 0

proving the result. Proposition 12.3 has a converse due to Kotani.

77

LYAPOUNOV EXPONENT AND SPECTRUM

Proposition 12.5. (Kotani) Let I ⊂ R be an interval s.t. mes(I ∩ [L(E) = 0]) > 0.  Then xac ∩ I = φ, x a.s. Corollary 12.6.  x  mes [L(E) = 0]\ = 0, x a.s. ac

Proof of Proposition 12.5. We follow basically the exposition in [Si]. Let Im z > 0. Define u± = (H± − z)−1 e0 ∈ 2 (Z± ∪ {0}) Here H+ (resp. H− ) is the restriction of H to [0, ∞[ (resp. ] − ∞, 0]). It follows that   (H − z) u− (0)u+ + u+ (0)u− − u+ (0)u− (0)e0   = [u− (0)u+ (1) + u+ (0)u− (−1) + v(x) − z u+ (0)u− (0)]e0 and hence u+ (0)u− (0)

  = G(z, x)(0, 0)[u− (0)u+ (1) + u+ (0)u− (−1) + v(x) − z u+ (0)u− (0)]

Since u± (0) = (H± − z)−1 e0 , e0  = 0 we may divide by u+ (0).u− (0) and obtain, denoting m± (x, z) = −

u± (±1) u± (0)

that G(z, x)(0, 0)−1 = −m+ (x, z) − m− (x, z) + v(x) − z

(12.7)

  Denote H+ (resp. H− ) the restriction of H to ]0, ∞[ (resp. ] − ∞, 0[). Since      − z) u+ (n)en = [ v(x + ω) − z u+ (1) + u+ (2)]e1 = −u+ (0)e1 (H+ n≥1

and hence  u+ (1) = − (H+ − z)−1 e1 , e1 u+ (0)  m+ (x, z) = (H+ (x) − z)−1 e1 , e1 

it follows that m± (x, z) are Herglotz functions of z. Let u + be the extension of u+ to Z satisfying (H − z) u+ = 0 (thus u + (n) = u+ (n) for n ≥ 0 but u + ∈ 2 ). Taking covariance considerations into account, it follows that (x)

(T x)

u + ∼ S u+ where

T x = x + ω and (Sξ)n = ξn−1

78

CHAPTER 12

Since 

v(x) − z) +

u + (1) + u + (0)

1 u+ (0) u+ (−1)

=0

we obtain therefore v(x) − z − m+ (x, z) −

1 m+ (T −1 x, z)

=0

(12.8)

Taking imaginary parts, it follows that Im z + Im m+ (x, z) =

Im m+ (T −1 x, z) |m+ (T −1 x, z)|2

2 log |m+ (T −1 x, z)| = log Im m+ (T −1 x, z) − log[Im z + Im m+ (x, z)] and integrating in x    2 log |m+ (x, z)|dx = − log 1 +

Im z Im m+ (x, z)

 dx

Applying the inequality log(1 + t) ≥ we get

t for t ≥ 0 1+t

 2

 log |m+ (x, z)|dx ≤ −

Im z dx Im m+ (x, z) + Im z

(12.9)

Write next un un un−1 u1 = · ··· u0 un−1 un−2 u0 n 

log |m+ (T j−1 x, z)| =

j=1

and

Hence



un u0 un+1 u0

uj = log un log u0 uj−1 j=1

n 



 = Mn (x, z)

1



u1 u0

 | + |uun+1 0|   u − log Mn (x, z) ≤ log un0 + un+1 u0

1 ≤ Mn (x, z)−1 



|un | |u0 |

Dividing by n and letting n → ∞, it follows that   1 log Mn (x, z)dx = −L(z) log |m+ (x, z)|dx ≥ − lim n

(12.10)

where L(z), z ∈ C is the subharmonic extension of the Lyapounov exponent satisfying L(z) < log(|z| + C).

79

LYAPOUNOV EXPONENT AND SPECTRUM

Thus, from (12.9) and (12.10), 

L(z) dx ≤2 Im m+ (x, z) + Im z Im z

(12.11)

By assumption, K = [L(E) = 0] ∩ I has positive measure. We may write on a complex neighborhood of I  L(z) = log |z − w|ρ(dw) where ρ is some positive measure. Thus L(E + iε) − L(E) ε

lim

ε→0

exists for almost all E, and hence lim

ε→0

L(E + iε) ε

exists for almost all E ∈ K. In fact, taking z = E + iε, ε > 0 in (12.11), one has that   dxdE 0 K T Im m+ (x, E + iε) + ε

(12.12)

Returning to (12.7) and taking imaginary parts, Im m+ (x, E + iε) + Im m− (x, E + iε) + ε =

Im G(E + iε, x)(0, 0) |G(E + iε, x)(0, 0)|2

where Im m+ ≥ 0, Im m− ≥ 0. Therefore, (12.12) implies x a.s.  |G(E + iε, x)(0, 0)|2 0. Proposition 12.5 is proven. Reference [Si] B. Simon. Kotani theory for one dimensional stochastic Jacobi matrices, CMP 89 (1983), 227–234. Since from the covariance property H(x + ω) = S −1 H(x)S (S = shift) it follows that Spec H(x) = Spec H(x + ω)

80

CHAPTER 12

and hence Spec H = Spec H(x) =

x

=

x pp

∪

x sc

∪

x ac

does not depend on x. Notice that each of the components may depend on x, however. For instance, in the xAlmost Mathieu case H(x) = λ cos(x + nω)δnn + Δ with λ large and ωDC, pp = φ for an uncountable (measure 0) set of x-values. In what follows we continue to consider operators H(x) = λv(x + nω)δ nn + Δ, ωDC. Proposition 12.14. (Bourgain) Assume that L(E) > 0 for all energies E . Then Spec H has positive Lebesgue measure. We treat the 1-frequency case. There are some additional technicalities to deal with the multifrequency case we don’t want to go into here. The argument is also generalizable to other models. Lemma 12.15. Assume L(E) > c0 > 0 for all E . Let E : I → R be a continuous function on a subinterval I ⊂ [0, 1], |I| < 10 −3 , and N a large integer and 0 < κ < 1 satisfying the following conditions: log log

1 log N |I|

(12.16)

N 1 log log |I| κ For each x ∈ I , there is a vector ξ ∈ [ej |j| < N ], ξ = 1 s.t.    H(x) − E(x) ξ < κ log log

(12.16 )

(12.17)

Then mes E(I) > e−(log |I| ) N

C

(12.18)

Proof. The set E(I) is an interval [E0 − ε, E0 + ε], and our aim is to obtain a lower bound on ε. Fix A  N N  K = log (12.19) |I| with A a sufficiently large constant. Since L(E0 ) > c0 , we may apply the Green’s function estimate from Proposition σ 7.19 at scale K. Thus, except for x in a set of measure < e −K (σ > 0 some constant), one of the intervals Λ = Λ(x) = [1, K], [1, K − 1], [2, K], [2, K − 1] will satisfy |GΛ (E0 , x)(n1 , n2 )| < e−c|n1 −n2 |+K

1−

for n1 , n2 ∈ Λ

(12.20)

Paving an interval [−N, N ] ⊂ Λ1 ⊂ [−N − 1, N + 1] with size K intervals Λ satisfying (12.20) and applying the resolvent identity (see Lemma 10.33) gives the following:

81

LYAPOUNOV EXPONENT AND SPECTRUM

For all x outside a set Ω with mes Ω < N e−K

σ

there is an interval [−N, N ] ⊂ Λ1 = Λ1 (x) ⊂ [−N − 1, N + 1] satisfying in particular GΛ1 (E0 , x) < eK

(12.21)

Recalling (12.19), we may ensure that mes Ω < mes I. Take x ∈ I\Ω and ξ the corresponding vector satisfying (12.17). Since ξ ∈ [e j |j| < N ] and choice of Λ1   (HΛ1 − E0 )ξ = (H − E0 )ξ = H − E(x) ξ + 0(ε)    HΛ1 (x) − E0 ξ < κ + ε and from (12.21), e−K < κ + ε Recalling (12.16) and (12.19), e−K  κ, so that ε > 12 e−κ . This proves (12.18). Remark. Lemma 12.15 may be formulated simply on the basis of Green’s function assumptions, independently of the Lyapounov exponent. Lemma 12.22. Let I ⊂ [0, 1] be an interval and E(x) ∈ Spec H N (x) a continuous function on I . Assume again that for all x ∈ I there is ξ ∈ [ej |j| < N ], ξ = 1. s.t.   c  H(x) − E(x) ξ < e−N (12.23)

(where c > 0 is some constant). Let log N1  log N  log log N1

Then there is a system (I  , EI  )I  ∈J  where J  is a collection of at most N1C intervals I  ⊂ I satisfying the previous assumptions with N replaced by N 1 , and moreover,   1 mes EI  (I  ) > mes E(I) − (12.24) N1 Proof. We assume v, the potential, a trigonometric polynomial to avoid some  approximation arguments. Let (λs = λs (x) |s|≤N1 be a continuous parametrization of Spec HN1 (x), and denote ϕs = ϕs (x) the corresponding eigenfunctions (no regularity properties in x will be involved). Fix x ∈ I. If ξ is the vector from (12.24), write  ξ=

ξ, ϕs ϕs and HN ξ = H N 1 ξ =



λs ξ, ϕs ϕs

= Eξ + 0(e−N )  c =E

ξ, ϕs ϕs + 0(e−N ) c

82

CHAPTER 12

Thus



|λs − E|2 ξ, ϕs 2

1/2

< e−N

c

It follows that there is some |s| ≤ N1 s.t. 1 | ϕs , ξ| > √ N1 and |λs − E|
√1N by (12.25), the preceding implies that 1

min

η∈[ek | |k| e2N1

(12.27)

(the vector η above is simply obtained as Pj ϕs ). Define for |s| ≤ N1 and N21 < j < N1 the set Γs,j ⊂ I of x for which (12.26) and (12.27) hold. Thus  I= Γs,j s,j

Observe (since v was assumed to be a trigonometric polynomial) that ζ = λs (resp. λs − E) are continuous functions on I satisfying an equation of the form  cr (x)ζ r = 0 (12.28) ζd + r 0

(12.29)

is obtained in (x, ζ = λs (x)), where ζ satisfies (12.28). Therefore, condition (12.27) also leads to at most N1C components. We may thus take each Γsj as a union of at most N1C intervals I  . For I  ⊂ Γsj , let EI  = λs , for which by construction   C  H(x) − EI  (x) η < e−N1 inf η∈[ek |k| N0−C and c

min

ξ∈[ek | |k| e−(log N0 )

C

Starting from (I0 , E0 ), apply Lemma 12.22 with log log N0 log log N1 log N0 to get a system (I, EI )I∈J1 satisfying the properties of the lemma. Next, repeat to each I ∈ J1 to obtain (I, EI )I∈J2 etc. Thus  %  mes EI (I) I∈Js

> mes



%

 EI (I) −

I∈Js−1

1 Ns

> · · · > mes E0 (I0 ) −

> 12 mes E0 (I0 ) The property (12.23) implies that   c dist EI (x), Spec H < e−Ns if x ∈ I ∈ Js . Hence

  s I∈Js

EI (I) ⊂ Spec H

1 N1

− ··· −

1 Ns

84

CHAPTER 12

proving the result. Remarks. 1. It follows from Proposition 12.14 and the results on Lyapounov exponents that if Hλ (x) = λv(x + nω)δnn + Δ, and |λ| > λ0 , then mes Spec Hλ > 0. The proof also enables us to establish this fact more generally for lattice Schrödinger operators Hλ (x) = λv(x + nω)δnn + Sφ v real analytic on T and where Δ is replaced by a Toeplitz operator S φ with real analytic symbol φ on T. (In Lemma 12.15 we apply directly the Green’s function result Proposition 11.4–valid in this generality–without involving Lyapounov exponents). 2. Return to the Aubry duality described in Chapter 11. Thus, starting from Hλ (x) = λv(x + nω)δnn + Δ (12.30) (1-frequency case), we introduced λ H(x) = cos(x + nω)δnn + Sv 2 Analyzing the construction, we see that = 1 Spec H Spec H 2 > 0 for λ sufficiently small, it follows that Since, by previous remark, mes Spec H mes Hλ > 0 both for |λ| > λ0 and for |λ| < λ1 ; λ1 , λ0 > 0. 3. In particular, in the Almost Mathieu case Hλ (x) = λ cos(x + nω) + Δ, mes Spec Hλ > 0 for |λ| = 2 It is known that in this particular case (12.31) mes Spec Hλ = 2 2 − |λ| (see [L] for a discussion and references). 4. It is conjectured that for (12.30), λ = 0, v nonconstant real analytic on T (1-frequency case), and ωDC, Spec Hλ has an empty interior. This is known in the following cases. Let v(x) = cos x. In the perturbative regime (λ large or small), Spec Hλ is a Cantor set [Sin]. If ω is an irrational Liouville number, rational approximations enable us to show that Spec H λ has no interior points (see [B-S]). If λ = 2, then mes Spec Hλ = 0 (see [G-J-L-S]), and the property is clear. In this situation, the renormalization group analysis from [H-S] enables a more precise description of the Cantor set for ω’s with continued fraction expansion ω = a1 + 1 1 1 a2 + a3 +···

satisfying aj > C for all j ≥ 1 where C is a sufficiently large constant (see the remark in Chapter 8). Very recently it was shown by J. Puig that the Almost Mathieu operator has Cantor spectrum for all λ = 0 and ω diophantine (preprint 2003).

LYAPOUNOV EXPONENT AND SPECTRUM

85

References [B] J. Bourgain. On the spectrum of lattice Schrödinger operators with deterministic potential, J. Anal. Math. 87 (2002), 37–75. [L] Y. Last. Almost everything about the Almost Mathieu operator, Proc. XI, International Congress Math Phys., Paris 1994, pp. 366–372. International Press, Cambridge, England, 1995. [Sin] Y. Sinai. Anderson localization for one-dimensional difference Schrödinger operator with quasi-periodic potential, J. Stat. Phys. 46 (1987), 861–909. [B-S] J. Bellissard, B. Simon. Cantor spectrum for the Almost Mathieu equation, J. Funct. Anal. 48(3) (1982), 408–419. [G-J-L-S] A. Gordon, S. Jitomirskaya, Y. Last, B. Simon. Duality and singular continuous spectrum in the Almost Mathieu equation, Acta Math. 178 (1997), 169–183. [H-S] B. Helffer, J. Sjöstrand. Mémoires de la SMF 34 (1988) and 39 (1989).

Chapter Thirteen Point Spectrum in Multifrequency Models at Small Disorder Consider Schrödinger operators on Z Hλ (x) = λv(x + nω)δnn + Δ

(13.1)

where v is a nonconstant trigonometric polynomial on Td . We proved that if λ > λ0 (v) then typically pure point spectrum with localization occurs. If d = 1, then for λ < λ1 (v), one obtains purely (absolutely) continuous spectrum. These results are nonperturbative in the sense that λ0 , λ1 do not depend on ω (which is always assumed diophantine). It turns out that for d ≥ 2, we cannot expect such nonperturbative statements for the continuous spectrum at small disorder. Notice that the dual model of (13.1) is a lattice Schrödinger operator on the Zd -lattice λ (x) = cos(x + n.ω) + λ Sv (13.2) H 2 Localization results for operators of the form (13.2) were obtained in [C-D] but λ (x) satisfies Anderson localization (and Hλ has only are perturbative. Thus H continuous spectrum) x a.s. for λ < λ1 (v, ω). The following statement shows that the condition on λ does depend on ω if d ≥ 2. Theorem 13.3. Let v = v(x1 , x2 ) be a trigonometric polynomial on T2 with a nondegenerate local maximum. Then, for ω in a set of positive measure, H = v(nω)δnn + Δ has some point spectrum (with localized states) and mes( p,p ) > 0.   Thus in the context of (13.1) (d = 2), for any λ = 0, mes pp (Hλ (x)) > 0 for (x, ω) ∈ T4 in a set of positive measure. Notice that this does not also exclude the presence of continuous spectrum (in other ranges). In   energy  fact, one may produce examples with both spectral types =  φ, =  φ in different energy regions  (see [B2]). Also, if λ is small, pp ac the phenomenon is nontypical in the sense that for pp (Hλ ) to be nonvoid, ω has to be restricted to a set of small (but positive) measure. Remarks. 1. In Theorem 13.3, the frequency vector ω will be chosen small. Due to this fact, some care will be required when applying the methods developed earlier to produce localized states. 2. The point spectrum in Theorem 13.3 will appear at the edge of the spectrum. 3. Localization for Z2 -operators has been established in [BGS] for the general

88

CHAPTER 13

quasi-periodic class H(x) = v(x1 + n1 ω1 , x2 + n2 ω2 ) + εΔ (Δ = Laplacian on Z2 ), where v is an arbitrary real analytic potential on T2 (such that the partial maps v(x1 , ·) and v(·, x2 ) are nonconstant). The method used is different from [C-D] but also perturbative; i.e., ε < ε0 (v, ω). We will next summarize the main ideas in proving Theorem 13.3. In order to avoid some inessential technical matters, consider the example v(x1 , x2 ) = λ(cos x1 + cos x2 ) with λ > 0 small (the idea described below works in general). Thus   Hλ,ω (x) = λ cos(x1 + nω1 ) + cos(x2 + nω2 δnn + Δ

(13.4)

Clearly, max Spec H ≤ 2(1+λ), and if |ω| = |ω1|+|ω2 | is small, then max Spec H ≈ 2(1 + λ). First we will construct an energy interval I near 2(1 + λ) and finitely many tiny subintervals Iα ⊂ I (not depending on x) s.t.  L(E) > c(λ) > 0 for all E ∈ I\ Iα (13.5) To be more precise about size, fixing a large integer N0 , we take 1 ∼ log N0 log |ω|

(13.6)

and |I| > ρ(λ) 1 log log ∼ log N0 |Iα | < log N log(#{Iα }) ∼ 0 Next, we prove that



 (H) ∩ I > c1 (λ) > 0

(13.7)

so that necessarily (for N0 large enough)    mes (H) ∩ (I\ Iα ) > 0

(13.8)

mes

Also, invoking (13.5) and Proposition 12.3,      (I\ Iα ) = φ x a.s. H(x) ac

and hence mes

 pp

   H(x) H(x) > 0



sc

x a.s.

To get point spectrum, put x = 0 and restrict ω according to Remark (3) following Corollary 10.34.%This ensures in particular that any extended state of H λ,ω (0) with energy E ∈ I\ α Iα is localized (notice that the Iα intervals do depend on ω). Therefore,     (I\ Iα ) ∩ Hω (0) = φ c

POINT SPECTRUM IN MULTIFREQUENCY MODELS AT SMALL DISORDER

and by (13.8), mes



89



pp

 Hω (0) > 0

Property (13.7) will be established using the constructive approach from Chapter 11. Consider the restriction HN0 of H to [−N0 , N0 ]. We will show that    π Spec HN0 (0, − + s) ∩ I > δ  (13.9) mes 2 |s| 0 depending on λ) with eigenvectors ξ ∈ [ej |j| ≤ N0 ] that are exponentially small for |j| > 34 N0 . This, together with the Lyapounov exponent assumption (13.5), will permit us to prove (13.7) as Proposition 12.14 by iteration of Lemma 12.22.   As we mentioned, the fact that ω is small see (13.6) requires some care since the LDT for the fundamental matrix applies only at scales N  N0 . Next, we pass to details. Consider (13.4) with 0 < λ < 1 fixed (possibly small). Fix a large integer N0 , and let 1 102 N

< ω1 < 0

2 102 N

(13.10) 0

1 2 < ω2 < 2 2 N0 N0

(13.11)

Consider the matrix

  π A(s) = R[−N0 ,N0 ] H 0, − + s R[−N0 ,N0 ] 2    = λ(cos nω1 + sin(nω2 + s) δnn + Δ |n|,|n |≤N0

with s ∈ [0, δ]. Considering the vector ζ =

A(0)ζ, ζ = λ

 √ 0 2 + (1 − 10−6 )λ Since dA(s) = λ cos(s + nω2 )δnn = λ cos s.Id + 0 ds first-order eigenvalue variation implies that λ0 (s) >

λ for s ∈ [0, δ] 2



1 N0



90

CHAPTER 13

Hence λ0 ([0, δ]) = I where I is an interval satisfying |I| ∼ δλ and min E > 2 + (1 − 10−6 )λ E∈I

Denote ξ, ξ = 1 an eigenvector of A(s) with eigenvalue E > 2 + (1 − 10 −6 )λ. Our aim is to show that |ξn | is small for |n| near N0 . Observe that A(s) − E has diagonal elements   dn = λ cos nω1 + sin(nω2 + s) − E satisfying, since |nω1 |
E − λ cos nω1 + 0(λδ) > 2 + λ − 10−6 λ − λ(1 − 14 |nω1 |2 ) + 0(λδ) > 2 + λ4 |nω1 |2 − 10−6 λ + 0(λδ) Hence, for |n| > 12 N0 , λ − 10−6 λ + 0(δλ) > 2 + 10−6 λ 16.104 and a simple Neumann inversion argument shows that 3N0 |ξn | < e−cλN0 for |n| > 4 In particular, for s ∈ [0, δ]    π min  H 0, s − ) − λ0 (s) ξ < e−cλN0 2 ξ∈[ej |j| 2 +

(13.12)

ξ=1

Next, we turn our attention to the Lyapounov exponent L(E) of (13.4). Our C −N0C , such that aim is to extract from % I at most N0 subintervals Iα , |Iα | < e (13.5) holds on I\ Iα . Once this fact is established, we may find a subinterval [a, b] ⊂ [0, δ], b − a > N0−C , such that for s ∈ [a, b],  λ(s) ∈ Iα Hence

  L λ(s) > c0

Iteration of Lemma 12.22 enables one to prove that then   1 1 + + ··· mes λ([a, b])\Spec H < N1 N2 where log Ns log Ns+1 log log Ns Thus

   1 λ mes Spec H ∩ (I\ Iα ) > mes λ([a, b]) > (b − a) > 0 2 2

(13.13)

91

POINT SPECTRUM IN MULTIFREQUENCY MODELS AT SMALL DISORDER

which is (13.8). Properties (13.5) and (13.8) are thus obtained, which, as explained earlier, enables us to deduce Theorem 13.3. An important point to notice is that when proving (13.13), use of the LDT to obtain the required Green’s function estimates only appears at scales N with > log N  log N and hence compatible with ω as in (13.10) and (13.11). log N ∼ 1 0 Proof of (13.5). Assume E > 2 + (1 − 10−6 )λ. The diagonal of H(x) − E is given by   dn (x) = λ cos(x1 + nω1 ) + cos(x2 + nω2 ) − E and hence, restricting |n − n ¯ | < N0 ,   λ |dn (x)| > |E − λ cos(x1 + n ¯ ω1 ) + cos(x2 + n ¯ ω2 ) | − 10 λ (13.14) >2 + 2 provided cos(x1 + n ¯ ω1 ) + cos(x2 + n ¯ ω2 ) ≤ 0 Denote N00 = N010 , say, and consider for fixed x ∈ T2 the orbit x + nω, 1 ≤ n ≤ N00 . From (13.10) and (13.11) it is clear that there is a collection of N 0 -intervals Λα in [1, N00 ] for which (13.14) holds; thus 

|GΛα (E, x)(n, n )| < λ−1 e−cλ|n−n | for n, n ∈ Λα

(13.15)

and satisfying 

1 |J| 10 whenever J ⊂ [1, N00 ] is an interval of size |J| > 100N0. Moreover, assume that E satisfies   1/2 dist E, Spec HΛ (x) > e−N0 |Λα ∩ J| >

(13.16)

(13.17)

and hence 1/2

GΛ (E, x) < eN0

for any subinterval Λ ⊂ [1, N00 ], |Λ| > N0 . It may then be shown using the resolvent identity (as in Lemma 10.33) that |G[0,N00 ] (E, x)(n, n )| < e10N0 and 

|G[0,N00 ] (E, x)(n, n )| < e−c(λ)|n−n | for n, n ∈ [0, N00 ], |n − n | >

1 N00 10

(see Lemma 13.23 below). Consequently, | det[HN00 (x) − E]|−1 = |G[0,N00 ] (E, x)(0, N00 )| < e−c(λ)N00 and thus 1 log MN00 (x, E) > c(λ) N00

(13.18)

92

CHAPTER 13

From condition (13.17), it is clear that (13.18) will hold for all (x, E) ∈ T2 × I 1/2 3 −N0 e . A Fubini argument therefore gives a except for a set of measure < N00 subset E of I such that 1

1/2

mes E < e− 2 N0

and if E ∈ E, then (13.18) holds for all x in a set of measure > 12 . Consequently, 1 c(λ) for E ∈ I\E 2 Starting from scale N00 , the proof of Proposition 7.2 enables us to show that LN00 (E) >

−1/2

|L(E) − LN00 (E)| < N00

(13.19)

Hence L(E) >

1 c(λ) for E ∈ I\E 3

(13.20)

Notice again that at scales N > N00 = N010 , the LDT for N1 log MN (x, E) applies. C It remains to obtain E as a union of at most N00 = N010C intervals. This is a consequence of the semialgebraic nature of the condition. One may define E as the set of E ∈ I such that 1/4 mes [x ∈ [0, 1]2 MN00 (x, E) < ec(λ)N00 ] > e−N0 (13.21) The set

A = [(x, E) ∈ [0, 1]2 × I MN00 (x, E)2HS < e2c(λ)N00 ]

is clearly semialgebraic of degree < CN00 . If (13.21) holds, there is some y ∈ [0, 1]2 such that (x, E) ∈ A for all x with 1/4 |x − y| < e−10N0 . Consider the set B ⊂ [0, 1]2 × [0, 1]2 × I of elements (x, y, E) for which $ 1/4 |x − y| < e−10N0 (x, E) ∈ A which is semialgebraic of degree < CN00 . Define E1 = ProjE C( Projy,E B)

(13.22)

It follows from the preceding that E ⊂ E1 and, also for E ∈ E1 1/4 mes [x ∈ [0, 1]2 MN00 (x, E)HS < ec(λ)N00 ] > e−10N0 Thus 1/2

1/4

3 −N0 e e10N0 mes E1 < N00 C and E1 is a union of at most N00 intervals. This proves (13.5).

1

1/2

< e− 2 N0

POINT SPECTRUM IN MULTIFREQUENCY MODELS AT SMALL DISORDER

93

Lemma 13.23. Let A = vn δnn + Δ be an N × N matrix such that |vn | < C . Assume there is a disjoint collection {Iα } of size M intervals in [1, N ] (M N ) s.t.  |Iα | ≥ cN

and for each α (RIα ARIα )−1  < eM

1−

(13.24)

and 

|(RIα ARIα )−1 (n, n )| < e−c|n−n | for |n − n | >

M 10

(13.25)

Assume further that for any interval I ⊂ [1, N ] (RI ARI )−1  < eM

1/2

(13.26)

holds. Then 



|A−1 (n, n )| < e10M−c d(n,n )

where for n < n d(n, n ) ≡

 [n, n ] ∩ Iα |

(13.27)

(13.28)

α

Proof. It is again an application of the resolvent identity. Denote J = [1, N ]. Assume that k1 < k2 ∈ J and d(k1 , k2 ) > 10M Case 1. k1 ∈ Iα for some α and M 5 It follows from the resolvent identity applied to the decomposition J = Iα ∪(J\Iα ) that  |A−1 (k1 , k2 )| ≤ |(RIα ARIα )−1 (k1 , k3 )| |A−1 (k4 , k2 )| dist (k1 , ∂Iα ) >

k3 ∈∂Iα ,k4 ∈J\Iα |k3 −k4 |=1

≤ ≤



k3 ∈∂Iα ,k4 ∈J\Iα |k3 −k4 |=1

M 5

≤e

max

≤|k1



e− 2 |k1 −k | |A−1 (k  , k2 )| c

−k |≤M

c − 10 M

e−c|k1 −k3 | |A−1 (k4 , k2 )|

max

|k1 −k |≤M

|A−1 (k  , k2 )|

(13.29)

by (13.25). Case 2. Assume Iα = [aα , aα + M ] and Iβ = [aβ , aβ + M ] consecutive intervals s.t.   M M , aβ + k1 ∈ aα + M − 5 5

94

CHAPTER 13

Put

  M M , aβ + I = aα + M − 4 4

Apply the resolvent identity to the decomposition J = I ∪ (J\I) to obtain  |A−1 (k1 , k2 )| ≤ |(RI ARI )−1 (k1 , k3 )| |A−1 (k4 , k2 )| k3 ∈I,k4 ∈J\I | k3 −k4 |=1

≤ e2M

1/2

max

k4 ∈I,dist (k4 ,I)=1

|A−1 (k4 , k2 )|

Clearly, k4 ∈ Iα ∪ Iβ , and if k4 ∈ Iβ , say, then dist (k4 , ∂Iβ ) > the Case 1 estimate (13.29), |A−1 (k4 , k2 )| ≤ e− 10 M c

max

|k4 −k |≤M

(13.30) M 5 .

Thus, from

|A−1 (k  , k2 )|

and substitution in (13.30) yields |A−1 (k1 , k2 )| < e− 11 M c

max

d(k ,k2 )>d(k1 ,k2 )−2M

|A−1 (k  , k2 )|

(13.31)

We use here the fact that d(k1 , k2 ) ≤ d(k1 , k4 ) + |k4 − k  | + d(k  , k2 ) M M + d(aα + M, aβ ) + + 1 + M + d(k  , k2 ) ≤ 5 4 and d(aα + M, aβ ) = 0, by definition (13.28). Thus estimate (13.31) holds in both Cases 1 and 2. Straightforward iteration then permits us to establish (13.27) for some constant c . Remark. In [B2], examples are produced of the form Hλ1 ,λ2 ,ω = (λ1 cos nω1 + λ2 cos nω2 )δnn + Δ  suchthat  for ω = (ω1 , ω2 ) in a set of (small) positive measure, ac (H) = φ and mes pp (H) > 0. Again here the frequency vector ω is chosen to be small. See [F-K] for results on quasi-periodic Schrödinger operators on R in this respect.

POINT SPECTRUM IN MULTIFREQUENCY MODELS AT SMALL DISORDER

95

References [B1, 2] J. Bourgain. On the spectrum of lattice Schrödinger operators with deterministic potential, I, II, J. Analyse Math. 87 (2002), 37–75, and 88 (2002), 221–254. [BGS] J. Bourgain, M. Goldstein, W. Schlag. Anderson localization for Schrödinger operators on Z2 with quasi-periodic potential, Acta Math. 188 (2002), 41–86. [C-Di] V. Chulaevsky, E. Dinaburg. Methods of KAM theory for long-range quasi-periodic operators on Zn . Pure point spectrum, Comm. Math. Physics 153(3) (1993), 559–557. [F-K] A. Fedorov, F. Klopp. Transition d’Anderson pour des opérateurs de Schrödinger quasiperiodiques en dimension 1, Séminaire: Equations aux Dérivées Partielles, 1998–1999, Expo. IV.

Chapter Fourteen A Matrix-Valued Cartan-Type Theorem The main result of this chapter is as follows: Proposition 14.1. Let A(σ) be a self-adjoint N × N matrix function of a real parameter σ ∈ [−δ, δ], satisfying the following conditions (i) A(σ) is real analytic in σ , and there is a holomorphic extension to a strip |Re z| < δ, |Im z| < γ

(14.2)

A(z) < B1

(14.3)

satisfying

(ii) For each σ ∈ [−δ, δ], there is a subset Λ ⊂ [1, N ] s.t. |Λ| < M

(14.4)

(R[1,N ]\Λ A(σ)R[1,N ]\Λ )−1  < B2

(14.5)

and

(iii)

mes [σ ∈ [−δ, δ] A(σ)−1  > B3 ] < 10−3 γ(1+B1 )−1 (1+B2 )−1 (14.6)

Then, letting κ < (1 + B1 + B2 )−10M

we have     c log κ−1 δ δ 1 − M. log(M −1 +B1 +B2 +B3 ) A(σ) < e mes σ ∈ − ,  > 2 2 κ

(14.7)

Proof. Denote 

 δ

δ1 = 10−2 γ(1 + B1 )−1 (1 + B2 )−1

Fix σ0 ∈ − δ2 , 2 . If z ∈ C, |z − σ0 | < δ1 , it follows that 1 (1 + B2 )−1 50 and hence, if Λ ⊂ [1, N ] is the index set associated to σ0 , we obtain from (14.5) (and a standard Neumann series argument) A(z) − A(σ0 ) ≤ 2B1 γ −1 δ1
B3−M Consequently, denoting a =

σ1 −σ0 δ1 , |a|


−M log B3 It follows from Jensen’s inequality that for 0 < ρ < 12 ,  u(a + ρeiθ )dθ ≥ u(a) and hence

 |z−a|< 12

u(z) > −M log B3

(14.15)

99

A MATRIX-VALUED CARTAN-TYPE THEOREM

Taking also (14.14) into account, it follows that < M log(M + B + B + B ) uL1 [|z−a|< 12 ] ∼ 1 2 3

From the Riesz-representation theorem,  1 1 u(z) = log |z − w|μ(dw) + 0(1) for |z − a| < M log(M + B1 + B2 + B3 ) 4 with μ ∈ M+ (|z − a| < 12 ), μ < C, and in particular, < M log(M + B + B + B ) uBMO(a− 14 ,a+ 14 ) ∼ 1 2 3

John-Nirenberg’s estimate implies that mes [σ |σ − σ1 | < 14 δ1 and | det B(σ)| < κ]   = δ1 mes x |x − a| < 14 and u(x) < − log κ1   c log κ−1 < δ1 exp − M log(M+B 1 +B2 +B3 ) Hence

  1 mes σ |σ − σ0 | < δ1 and | det B(σ)| < κ 8   log κ−1 < δ1 exp − c M log(M + B1 + B2 + B3 )

If |σ − σ0 |
κ 2 2 M log(M + B1 + B2 + B3 ) δ1 8 ,

which proves Proposition 14.1. Remarks. 1. In application, log κ1 = o(N ), and to obtain a nontrivial estimate from (14.7), we thus need to assume that M < N ρ B1 , B2 , B3 < eN

ρ

where ρ, ρ ≥ 0 satisfy ρ + ρ < 1. 2. In the previous argument we could have alternatively invoked Cartan’s result directly, once (14.14) and (14.15) established the (scalar) subharmonic function u. 3. Proposition 14.1 easily generalizes to matrix-valued functions A(σ) depending real analytically on a multiparameter σ. For instance, if σ = (σ1 , σ2 ), estimate (14.7) should be replaced by 1/2    log κ−1 −1 −1 mes [σ A(σ)  > κ ] < exp − c M log(M + B1 + B2 + B3 ) (14.17)

100

CHAPTER 14

4. Some comments on conditions (ii) and (iii) in Proposition 14.1. We always assume A(σ) with exponential off-diagonal decay 

|A(σ)(n, n )| < e−c0 |n−n |

(14.18)

Let N0 N 1 N be 2 scales. For given σ, define J (σ) as the collection of size-N 0 intervals I ⊂ [1, N ] s.t., denoting AI = RI ARI 1−

AI (σ)−1  < eN0

(14.19)

and N0 (14.20) 10 Assume that there are at most M < NN1 intervals I, |I| = N0 s.t. I is not in J (σ). Let {Iα } be a partition of [1, N ] in N0 -intervals and Λ the union of the Iα for which there is an N0 -interval I ⊂ [1, N ], I ∩ Iα = φ and I ∈ J (σ). Thus (14.4) holds, and from (14.18) through (14.20) and the resolvent identity, (14.5) follows with B2 = e N 0 . To establish (iii), we proceed as follows. Assume that it is established that for any N1 -interval J ⊂ [1, N ] the properties 

|AI (σ)−1 (n, n )| < e−c|n−n | for n, n ∈ I, |n − n | >

1−

AJ (σ)−1  < eN1

(14.21)

and 1 N1 (14.22) 10 hold, except for σ in a set of measure less than 10−3 N −1 γB1−1 e−N0 . Covering then [1, N ] by size-N1 intervals, another application of the resolvent identity implies (14.6) with B3 = eN1 . In this setting, estimate (14.7) becomes   δ δ   K mes σ ∈ − , A(σ)−1  > eK < e−c M.N1 (14.23) 2 2 5. The conclusion (14.7) in Proposition 14.1 only involves a bound on the inverse A(σ)−1 , while a multiscale inductive argument also requires off-diagonal decay estimates, (see (14.19) through (14.22)). This % is achieved as follows: We assume (14.18). Fix σ, and let Λ = Iα , |Iα | = N0 , the set introduced in the preceding remark. Assume that for a fixed 0 < ρ < 1 

|AJ (σ)−1 (n, n )| < e−c|n−n | for n, n ∈ J, |n − n | >

|Λ| = M < N ρ and N0 = N , N1 = N ρ0

ρ1

with ρ 0 , ρ1 ≤

1 (1 − ρ)2 10

Define ¯ = N 1−ρ 4 N

(14.24)

101

A MATRIX-VALUED CARTAN-TYPE THEOREM

¯ -interval and J ∩ Λ = φ, it follows from definition of Λ and If J ⊂ [1, N ] is an N the resolvent identity that AJ (σ)−1  < eN0

(14.25)

and 1 ¯ N 10



|AJ (σ)−1 (n, n )| < e−c|n−n | for n, n ∈ J, |n − n | >

(14.26)

¯ -interval “good.” Clearly, there are at most N ρ disjoint bad intervals. Call such N Assume in addition to (14.24) that |Λ ∩ J  | < |J  |ρ 

(14.27)



¯. also holds for any interval J ⊂ [1, N ] of size |J | = L > N To each of these intervals J  , apply Proposition 14.1 with N replaced by L > 1−ρ N 4 . By (14.27), condition (14.4) holds with M = L ρ and (14.5) with B2 = eN0 . Condition (14.6) is established as before with B3 = eN1 . Estimate (14.23) then implies that 1+ρ   L 2 1+ρ −c ρ ρ  δ δ  2 L ·N 1 −1 L | AJ  (σ)  > e N measure at most N 2 e−cN ¯

1−ρ 10

< e−N

10−2 (1−ρ)2

(14.29)

We then may apply the following lemma, which is a variant of Lemma 13.23, to conclude that for these σ, 

|A(σ)−1 (n, n )| < e−(c−)|n−n | for |n − n | >

N 10

(14.30)

Lemma 14.31. Assume A an N × N matrix satisfying 

|An,n | < e−c0 |n−n |

(14.32)

¯ = N τ for some 0 < τ < 1. Assume that for any interval J  ⊂ [1, N ], |J  | = Let N ¯, L≥N L A−1 J  < e

b

¯ -interval J “good”if, moreover, where 0 < b < 1, τ + b < 1. Call an N ¯ N  −c|n−n | |A−1 for n, n ∈ J, |n − n | > J (n, n )| < e 10

(14.33)

(14.34)

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CHAPTER 14

1 ¯ -intervals. (take 0 < c < 10 c0 ). Assume that there are at most N b disjoint bad N Under these assumptions 



|A−1 (n, n )| < e−c |n−n | for |n − n | >

N 10

(14.35)

where c = c − N −κ , κ = κ(τ, b) > 0. 1+ρ We then apply Lemma 14.31 with A = A(σ), b = max(ρ, 1+ρ 2 ) = 2 ,τ = 1−ρ 4 . The size of the exceptional σ-set at scale N is bounded by (14.29). With such estimate at scale N1 , satisfying (14.6) according to the discussion in Remark (4) 10−3 (1−ρ)2 . Hence, for the scales N0 , N1 , one may will require us to take N0 < N1 take N0 = N 10

−3

a(1−ρ)4

and N1 = N a(1−ρ)

2

(14.36)

a parameter. The main differences between Lemmas 13.23 and with 0 < a < 14.31 is that the off-diagonal of A is not given by Δ but assumed to satisfy a more general assumption (14.32) (this is a minor technical point). Also, some additional care is needed to preserve essentially the constant c in (14.35). See [B] for details. 6. The method described in this chapter does have applications to lattice Schrödinger operators on ZD , D > 1. See, for instance, [BSG] for results on perturbative quasiperiodic localization on the Z2 -lattice obtained along these lines. 1 10

A MATRIX-VALUED CARTAN-TYPE THEOREM

103

References [B] J. Bourgain. Estimates on Green’s functions, localization and the quantum kicked rotor model, Annals of Math. 156(1) (2002), 249–294. [BGS] J. Bourgain, M. Goldstein, W. Schlag. See Chapter 13.

Chapter Fifteen Application to Jacobi Matrices Associated with Skew Shifts We consider 1D lattice Schrödinger operators H(x), x ∈ Td associated with a skew shift transformation T : Td → Td , and thus Hm+1,n+1 (x) = Hm,n (T x). To simplify matters, let d = 2 and T : T2 → T2 : (x1 , x2 ) → (x1 + x2 , x2 + ω) (the method applies equally well to higher-dimensional skew shift extensions). We always assume ω satisfying a DC. To avoid additional parameters, assume, say, k.ω > c|k|−2 for k ∈ Z\{0}

(15.0)

H(x) will be given by H(x) = V (T n x)δnn + δΔ

(15.1)

where V is a real nonconstant trigonometric polynomial on T . More generally, we will consider H(x) of the form 2

Hnn (x) = V (T n x)

(15.2)

Hmn (x) = φm−n (T m x) + φn−m (T n x)

(15.3)

V is a real nonconstant trigonometric polynomial on T2

(15.4)

φk is a trigonometric polynomial of degree < |k|C1

(15.5)

φk ∞ < δ e−|k|

(15.6)

and for m = n where again

Our purpose is to obtain Green’s function estimates and (dynamical) localization results. The model (15.1) also may be treated by the transfer matrix approach, (see [BGS]), but no nonperturbative results are known so far in the skew shift case. The model (15.2) through (15.6) is of importance to our main application, which is the kicked rotor equation. This is the linear Schrödinger equation (1.4) with periodic time-dependent potential    ∂u ∂u ∂ 2u i δ(t − n) u = 0 (15.7) + a 2 + ib + κ cos x · ∂t ∂x ∂x n∈Z

106

CHAPTER 15

form It’s monodromy matrix W∗ , as we will see, turns out to be of the preceding  more precisely, W + W satisfies the description (15.2) through (15.6) . Obtaining the Green’s function bounds will be an application of the preceding chapter (this approach works in a much more general context than the fundamental matrix technique, but results are perturbative). Proving localization and dynamical localization will then be achieved following the same scheme as in Chapter 10 (for the shift model) based on semialgebraic set theory. Of course, the required facts about intersections of semialgebraic sets and skew shift orbits will have to be established. Our first goal is thus to prove Proposition 15.8. Let H be given by (15.2) through (15.6) with δ in (15.6) suffi ciently small. Then, for all suffi ciently large N and energy E , we have that |GN (E, x)| < eN

1−

(15.9)

and 

1

|GN (E, x)(n, n )| < e− 100 |n−n | for |n − n | >

N 10

(15.10)

except for x ∈ ΩN (E) ⊂ T2 satisfying mes ΩN (E) < e−N

σ

(15.11)

(for some σ > 0 that will be specifi ed later). The proof is multiscale and perturbative. To start off, take N0 a large integer. For δ = δ(N0 ) small enough, one then has that (for any 0 < θ < 1) 1



|GN0 (E, x)(n, n )| < eN0 − 2 |n−n | for n, n ∈ [1, N0 ] θ

(15.12)

−cN0θ

. except for x in ΩN0 ,θ (E) with mes ΩN0 ,θ (E) < e Indeed, from assumption (15.4) and Lojasiewicz’ inequality it follows that < γc mes [x ∈ T2 | |V (x) − E| < γ] ∼

for all γ > 0 and where c > 0 is a constant depending on V (not on E). Thus mes [x ∈ T2 | min |V (T n x) − E| < γ] < N0 γ c 0≤n≤N0

If min0≤n≤N0 |V (T x) − E| > γ > Cδ, we get by Neumann expansion and (15.6) n

|G[0,N0 ] (E, x)(m, n)| ≤     δ s −1 γ 1+ γ s≥1





e

−(|m−n1 |+···+|ns−1 −n|)

1≤n1 ,...,ns−1 ≤N0

1

< γ −1 e− 2 |m−n| Putting γ < e−N0 and δ = e−N0 , (15.12) follows. The inductive step is achieved using the analysis from Chapter 14. The parameter σ = (x1 , x2 ) ∈ T2 and A(x) = HN (x) − E has entries given by trigonometric polynomials of degree < N 1+C1 . This follows from (15.5) and the fact that θ

T n x = (x1 + nx2 +

n(n − 1) ω, x2 + nω) 2

APPLICATION TO JACOBI MATRICES ASSOCIATED WITH SKEW SHIFTS

107

Thus, in conditions (14.2) and (14.3) on the analytic extension, take γ = N −C1 −2 , B1 = 1 C. Following Chapter 14, consider a scale N0 , log N0 < 10 log N , log N0 ∼ log N and to be specified later. Following Remarks (4) and (5) in Chapter 14, the issue is to establish (14.24) through (14.27) for some fixed, ρ < 1. Thus we need to show there is 0 < ρ < 1 such that |Λ ∩ J| < |J|ρ

(15.13)

holds for any subinterval J ⊂ [1, N ] of size |J| = L > N complement of sites n0 s.t., if I = [n0 , n0 + N0 ]

1−ρ 4

and where Λ is the

1−

AI (x)−1  < eN0

(15.14)

 N0 (15.15) |AI (x)−1 (n, n )| < e−c|n−n | for n, n ∈ I, |n − n | > 10   condition (14.19) and (14.20) . Here x ∈ T2 is fixed, and the preceding should hold uniformly. Also,

AI (x) = HN0 (T n0 x) − E Letting ΩN0 (E) be the complement of the set of x ∈ T2 for which 1−

GN0 (E, x) < eN0

(15.16)

and 1



|GN0 (E, x)(n, n )| < e− 100 |n−n | for |n − n | >

N0 10

(15.17)

we need thus to estimate

#{n ∈ J T n x ∈ ΩN0 (E) (mod 1)}

(15.18)

From (15.11) and the induction hypothesis, we have mes ΩN0 (E) < e−N0

σ

(15.19)

Expressing again GN0 (E, x) as a ratio of determinants, ΩN0 (E) clearly may be viewed as a semialgebraic set of degree at most N03+C1 . σ 1 Fix ε > e− 10 N0 (in its later choice, log 1ε ∼ log N , and this condition will obviously hold). It follows from the uniformization theorem for semialgebraic sets (Theorem 9.4) that ∂ΩN0 (E); hence ΩN0 (E) ⊂ T2 may be covered by at most N0C ε−1 discs of radius ε. Given a, x ∈ T2 , estimate next #{n = 1, . . . , L T n x − a < ε} (15.20) where n(n − 1) ω − a1  + x2 + nω − a2  2 (  refers to the distance on T or T2 ). Lemma 15.21. Assume that ω satisfi es (15.0), and let ε > L−1/10 . Then #{n = 1, . . . , L T n x − a < ε} < Cε2 L T nx − a = x1 + nx2 +

108

CHAPTER 15

Proof. Majorizing χΔ(a,ε) by a trigonometric polynomial of degree < may be bounded by the expression L     2πik.T n x 2 ε L+ e k∈Z2 ,0 L−1/10 . Taking ε = L− 10 , L > N020C , we therefore obtain (15.13) 1−ρ 1 4 with ρ = 19 becomes L > N 80 . We let N0 = 20 . The condition L > N −3 2   −10 −4 10 (1−ρ) see (14.36) . From (14.29), N 10 /C , N1 = N 10 /C so that N0 < N1 10−2 (1−ρ)2

10−5

mes ΩN (E) < e−N < e−N and we may let σ = 10−5 in (15.11). This proves Proposition 15.8. The proof of localization and dynamical localization then follows the same strategy as explained in Chapter 10. The key ingredients are Proposition 15.8 and the following fact: Lemma 15.26. Let S ∈ T3 be a semialgebraic set of degree B s.t. mes S < e−B for σ > 0 σ

(15.27)

APPLICATION TO JACOBI MATRICES ASSOCIATED WITH SKEW SHIFTS

109

Let M be an integer satisfying log log M log B log M

Thus, for any fi xed x0 ∈ T , mes[ω ∈ T (ω, Tωj x0 ) ∈ S for some j ∼ M ] < M −c

(15.28)

2

(15.29)

for some c > 0. (Tω denotes the skew shift with frequency ω.) Proof. For notational reasons, denote (x1 , x2 ) ∈ T2 by (x, y). The issue is thus the intersection of S ⊂ [0, 1]3 and sets j(j − 1) ω, y0 + jω)|ω ∈ [0, 1]} {(ω, x0 + jy0 + (15.30) 2 where x0 + jy0 + j(j−1) ω and y0 + jω are considered (mod 1). 2 σ We first apply Lemma 9.9 to be set S (with η = e−B ) in the product [0, 1]×[0, 1]2 2 (thus ω ∈ [0, 1], (x, y) ∈ [0, 1] ). Take ε = M −1+ and consider the decomposition S = S1 ∪ S2 satisfying (9.10) and (9.11). Since Projω S1 has measure < B C M −1+ = M −1+ , restriction of ω permit us to replace S by S2 , satisfying mes 2 (S2 ∩ L) < B C ε−1 η 1/3 < η 1/4 ε whenever L is a plane satisfying |ProjL (eω )| < 100 . Fixing j, notice next that (15.30) considered as a subset of [0, 1]3 lies in the union of the parallel planes   m + y0 y S(j) − e0 (m ∈ Z, |m| < M ) = S = ω = m m j j

. .. . ... .. .. .. ... ... ... ... ... .. .. ... ... .... .. .. . .... .... .... ... ... ... .. ... ... ... ... ... ... .. .. ... ... ... .. .. .. .... ..... ..... .. .. ... .. .. ... ... ... ... .. .. .. ... ... ... ... ... .. .... .... ..... .. ... ... .. .. .. ... ... ... .. .. .. ... ... ... .. ... ... .... .... .... . ... ... ... .. .. .. ... ... ... .. ... ... ... ... ... ... .. .. .... .... .... ... ... ... .. ... ... ... ... ... ... .. .. ... ... ... .. .. .. .... .... .... .. .. .. ... ... ... ... ... ... .. .. .. ... ... ... .. .. ... ... .... .... . . . . ... ... ... .. ... ... ... ... ... .. .. ... ... ... ... ... .. ... ... ...

. ...... .. ..

y

Sm

.......... .....

ω

−y0 +m j

< M −1 ε. Hence, for each Thus Sm ⊥ ζj //(1, 0, − j1 ), j ∼ M and |ζj − e0 | ∼ m,

mes (S2 ∩ Sm ) < η 1/4

(15.31)

110

CHAPTER 15

Fixing m, consider the semialgebraic set S2 ∩ Sm and its intersection with the parallel lines (j)

Lm,m = Lm,m"

= Sm ∩ y =

...... ... ..

x

Sm

2 j−1 x

−

.. .. .. .. ... ... ... .. .. ... ... ... ... .. .. ... ... .... .... . . ... ... ... ... .. .. ... ... ... .. .. .. ... ... ... .. ... .. ... .... . ..... ... ... ... ... .. .. ... ... ... .. .. .. ... ... ... .. ... .. .... .... . ..... ... ... ... .. .. .. ... ... ... .. .. .. ... ... ... .. ... .. ... .... . ..... . . ... ... ... .. .. .. ... ... ... ... .. .. ... ... ... .. .. .. .... .... .... ... ... ... .. .. ... ... ... ... ... ... .. ... ... ... .. .. .. .... .... .... ... ... ... .. .. ... ... ... ... .. ... .. ... ... ... ... .. .. .... .... .... .. ... ... ... .. .. ... ... ... .. .. ...

2 j−1 (x0

+

j+1 2 y0

# − m ) (m ∈ Z, |m | < M )

Lm,m

........... ......

y  Sm ∩ [x = 0]

Apply again Lemma 9.9 in the Sm -plane (n = 1) considering the set S2 ∩ Sm (x0 = y, x1 = x), and let ε = M −1+ . The set S2 ∩ Sm decomposes as   S2 ∩ S m = S m ∪ Sm (depending on j)

where  ProjSm ∩[x=0](Sm ) is a union of at most B C intervals of measure at most B C M−1+ (15.32) and, for all m , by (15.31)  mes (Sm ∩ Lmm ) < B C M mes (S2 ∩ Sm )1/2 < B C M η 1/8

(15.33)



Summing (15.33) over j, m, m , the collected contribution in the ω = parameter is thus less than 1 M 3 B C M η 1/8 < η 1/9 M2  Consider the contribution of the sets Sm . Thus    δj,m ≡ mesω Projω ProjSm ∩[x=0] (Sm ) < B C M −2+ (15.34) If the statement (15.29) fails, we have thus  δj,m > M 0−

(15.35)

j∼M |m| M 1− such that for each j ∈ J there are at least M 1− values of m satisfying  (j)  mes ProjSm ∩[x=0] (Sm ∩ Lmm ) > M −1+ |m | M 0− max 

(15.36)

m

  (j) For fixed j, Lm,m //ξj , ξj // 1, j(j−1) , j and |ξj | = 1. Since (15.36) holds for at 2 least M 1− values of m, considerations of semialgebraicity, in particular Bezout’s theorem, permit us to find an algebraic curve Γ in the (ω, y)-plane such that the cylindrical surface C (j) = Γ + tξj

(15.37)

intersects Sη1 in a set of (2D)-measure > M 0− . Here Sη1 denotes the η1-neighborhood of S, and log η −1 ∼ log η1−1 . Moreover, one may fix Γ for j in a set J  ⊂ J , with still |J  | > M 1− . Denoting again N (S, η1 ) the metric entropy numbers, it follows in particular from Theorem 9.4 that N (∂S, η1 ) < B C η1−2 and N (S, η1 ) < B C η1−2 On the other hand, it follows from the preceding that for j ∈ J  , N (Sη1 ∩ C (j) ) > M 0− η1−2 Therefore,





N (Cη(j) ∩ Cη(j1 ) ) > M 2− η1−2 1

(15.38)

j,j  ∼M 

Roughly, (15.38) means that for many pairs j, j  the cylinders C (j) , C (j ) have η1 neighborhoods with large intersections. But the vectors {ξ j } are not coplanar,  clearly leading to a contradiction we use here again (15.28) . This proves Lemma 15.26. Theorem 15.39. Consider a lattice operator Hω (x) associated to the skew shift T = Tω acting on T2 and of the form (15.2) through (15.6). Fix x0 ∈ T2 . Then for almost all ω satisfying a specifi ed DC and δ taken suffi ciently small in (15.6), Hω (x0 ) satisfi es Anderson localization and dynamical localization. Remarks. 1. The method described in Chapters 14 and 15 is a new perturbative approach to control certain Green’s functions developed in [B2] and [BGS2]. It does have a wide range of applications. 2. The particular case Hω (x) = v(Tω x)δnn + Δ

Tω x = (x1 + x2 , x2 + ω)

may be treated differently, following the method used for the ordinary shift (based on the transfer matrix formalism). But this approach, which we briefly describe below, also has failed so far to produce nonperturbative localization results. Thus we establish again an LDT for the transfer matrix using the methods from Chapters 4 through 7. This argument is more involved, however, than in the shift case. The method is closely related to arguments in Chapter 7 and is perturbative. The argument may be summarized as follows (see [BGS] for details):

112

CHAPTER 15

Assume that we dispose of inequalities at scale N0 σ 2 1 mes [x ∈ T log MN (x, E) − LN (E) > N0−σ ] < e−N0 N

(15.40)

for N = N0 , 2N0 and some σ > 0, where, say, LN0 (E) > 1

(15.41)

and 1 LN (E) (15.42) 100 0 At an initial scale, this may be ensured for v = λv0 (v0 = a nonconstant real analytic function on T2 ) by taking λ sufficiently large. Using (15.40) through (15.42), apply first the avalanche principle with LN0 (E) − L2N0 (E)
n Thus | log An · · · A1  +

n−1 

log Aj  −

j=2

n−1 

log Aj+1 Aj  | < C

j=1

n μ

and hence, letting N1 = nN0 , n−1 1 log MN1 (x, E) + 1  1 log MN0 (T jN0 x, E) N1 n N0 j=2

− n2

n−1  j=1

1 2N0

log M2N0 (T

jN0

x, E)
N1 ] < e−N1 N1 j=0 Hence

2

mes [x ∈ T

N −1 1 1  −c/2 u(T j x) − LN0 (E)| > N1 ]< N1 j=0

c/5

mes (T\Ω2 ) + e−N1

c/5

< 2e−N1

Returning to (15.44), we proved thus that 1 N0 −c/2 < −c/2 < (15.47) ∼ N1 N1 log MN1 (x, E)+LN0 (E)−2L2N0 (E) ∼ N1 +N1 1

1

c/5

except for x in a set of measure < e− 2 N0 + 10e−N1 < 2e− 2 N0 . The function u1 = u1 (x1 , x2 ) = N11 log MN1 (x, E) admits a pluri-subharmonic σ

σ

−c/2

extension bound by CN1 , and we apply Lemma 4.12 with ε0 ∼ N1 σ/2 σ 1 2e− 2 N0 , and B = N1 < N0 eN0 . Thus 1 −c log MN1 (x, E) − LN1 (E) > N1 9 ] < mes [x ∈ T2 N1 − 9c

exp −[N1

1

c 10

+ N1 e− 10 N0 ]−1 < e−N1 σ

, ε1 =

(15.48)

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CHAPTER 15

where −c/2

|LN1 (E) + LN0 (E) − 2L2N0 (E)| < 2N1

The same holds with N1 replaced by 2N1 . Thus (15.48) gives (15.40) at scale M = N1 , 2N1 , with σ =

c 10 .

(15.49) From (15.49),

−c/2

LN1 (E) > 2L2N0 (E) − LN0 (E) − 2N1 > LN0 (E) −

1 100 LN0 (E)

−c/2

− 2N1

(15.50)

and −c/2

|LN1 (E) − L2N1 (E)| < 4N1

(15.51)

3. A few comparisons between the 1D operators and

λ cos(x + nω)δnn + Δ

(15.52)

  n(n − 1) ω δnn + Δ λ cos x1 + nx2 + 2

(15.53)

(we assume ω diophantine and make statements x a.e.). While for the Almost Mathieu operator (15.52) the spectral type depends on λ (with transition at λ = 2) and the spectrum has a Cantor structure (at least proven in certain cases), the “conjectured” behavior for (15.53) is as follows (i) If λ = 0, then L(E) > 0 for all energies (ii) For all λ = 0, (15.41) has p.p. spectrum with Anderson localization (iii) There are no gaps in the spectrum. Based on the weakly mixing properties of the skew shift, the expected behavior is thus that of the random case. At this point, the known results are (iv) (15.53) satisfies Anderson localization for |λ| sufficiently large (depending on the DC for ω).  (v) For all λ = 0, there is a set of ω’s of positive measure for which mes pp > 0. Statement (iv) follows from Theorem 15.39 and as mentioned, [BGS] contains a different proof. Statement (v) at least exhibits some differences with the shift model. See [B1] for the proof, which has similarities with the argument in Chapter 9 for the multifrequency shift.

APPLICATION TO JACOBI MATRICES ASSOCIATED WITH SKEW SHIFTS

115

References [B1] J. Bourgain. Estimates on Green’s functions, localization and the quantum kicked rotor model, Annals of Math. 156(1) (2002), 249–294. [B2] J. Bourgain. On the spectrum of lattice Schrödinger operators with deterministic potential, J. Analyse Math. 87 (2002), 37–75 and 88 (2002), 221–254. [GBS] J. Bourgain, M. Goldstein, W. Schlag. Anderson localization for Schrödinger operators on Z with potentials given by the skew shift, Comm. Math. Phys. 220(3) (2001), 583–621. [BGS2] J. Bourgain, M. Goldstein, W. Schlag. Anderson localization on Z 2 with quasi-periodic potential, Acta Math. 188 (2002), 41–86.

Chapter Sixteen Application to the Kicked Rotor Problem We consider the time-dependent Schrödinger equation on T = R/Z ∂ 2 Ψ(t, x) ∂Ψ(t, x) ∂Ψ(t, x) =a + V (t, x)Ψ(t, x) + ib 2 ∂t ∂x ∂x with potential   δ(t − n) V (t, x) = κ cos 2πx i

(16.1)

(16.2)

n∈Z

corresponding to a periodic sequence of kicks. The monodromy operator W defined by W ψ(t, x) = Ψ(t + 1, x) (16.3)   2 = time-1 shift under the flow of (16.1) is a unitary operator on L (T). In the case of (16.1), W is given by W = Ua,b · W1,κ

(16.4)

where Ua,b d2

d

Ua,b = ei(a dx2 +ib dx )

(16.5)

W1,κ = multiplication operator by eiκ cos 2πx ≡ ρ(x)

(16.6)

and (see [Sin] and [Bel]). After passing to Fourier transform, Ua,b becomes a diagonal matrix Uab = e−i(4π

2

an2 +2πbn)

δmn

(16.7)

and W1,κ becomes a Toeplitz matrix W1,κ (m, n) = ρˆ(m − n) where one easily verifies that ρ(k)| < ρˆ(0) = 1 + 0(κ2 ) and |ˆ

√ −c(log 1 )|k| κ κe for k ∈ Z\{0}

(16.8)

(16.9)

Hence, for κ small, W1,κ is a perturbation of the identity with exponential offdiagonal decay. Defining H=

1 (W + W ∗ ) 2

(16.10)

118

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we get a self-adjoint operator 2 2 2 2 1 1 Hmn = e−i(4π am +2πbm) ρˆ(m − n) + ei(4π an +2πbn) ρˆ(n − m) (16.11) 2 2 that has the format considered in Chapter 15. Define 1 ρ(0) e−2πix + ρˆ(0) e2πix ) V (x, y) = (ˆ (16.12) 2 1 (16.13) φk (x, y) = ρˆ(k) e−2πix 2 √ by (16.9) satisfying conditions (15.4), (15.5), and (15.6) with γ = κ. Define ω = 4πa, x0 = 0, y0 = b + 2πa

(16.14)

and let T be the skew shift on T with frequency ω. Then   m(m − 1) m ω, y0 + mω T (x0 , y0 ) = my0 + 2   = mb + 2πm2 a, b + 2π(2m + 1)a 2

Hence, from (16.11),

  Hmm = V T m (0, y0 )

(16.15)

m

n

Hmn = φm−n (T (0, y0 )) + φn−m (T (0, y0 ))

(16.16)

and (15.2) and (15.3) hold (here T (0, y0 ) refers to its first coordinate). Thus Theorem 15.39 applies, and fixing initial (x0 , y0 ), localization holds for ω outside a set of small measure. By (16.14), it follows thus in particular that if the parameters (a, b) are restricted outside a set of small measure (→ 0 if κ → 0), then H and hence W have pure point spectrum with exponentially decaying eigenfunc tions {ϕα }. These eigenfunctions satisfy, moreover, the following property see (10.37) : 1/2  2 2 (1 + n )|ϕα (n)| | ψ, ϕα | < ∞ (16.17) m

α

n

assuming |ψn | < |n|−A for |n| → ∞ (for A a sufficiently large constant). Writing W ϕα = eiθα ϕα we have for r ∈ Z+ W rψ = and thus



eirθα ϕα , ψϕα

1/2 |n|2 |(W r ψ)(n)|2 1/2    ≤ | ϕα , ψ| |n|2 |ϕα (n)|2 0 d

ψ(t)H s tε for t → ∞ Thus, if diffusion to higher Fourier modes occurs when t → ∞, it necessarily happens slowly. Such slow diffusion may occur in H s for any s > 0 even if V is smooth and periodic in both x and t (see [B1]). Notice that in the kicked rotor result we did exploit the presence of parameters (a and b) to establish absence of diffusion.

122

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References [Bel] J. Bellissard. Noncommutative methods in semiclassical analysis, Springer LNM 1589 (1994), 1–64. [B1] J. Bourgain. Growth of Sobolev norms in linear Schrödinger equations with quasi-periodic potential, CMP 204 (1999), 207–247. [B2] J. Bourgain. On the growth of Sobolev norms in linear Schrödinger equations with smooth time potential, J. Analyse 72 (1999), 289–310. [CCFI] G. Casati, B. Chirikov, J. Ford, F. Izraelev. Lecture Notes in Physics, pp. 334–352, Springer-Verlag, Berlin, 1993. [CIS] B. Chirikov, F. Izraelev, D. Shepelyanskii. Dynamical stochasticity in classical and quantum mechanics, Math. Phys. Reviews 2 (1981), 209–267. [FGP] S. Fishman, D. Grempel, R. Prange. Chaos, quantum recurrences, and Anderson localization, Phys. Rev. Letters 49 (1982), 509–512. [Sin] Y. Sinai. Mathematical problems in the theory of quantum chaos, Springer LNM 1469 (1991), 41–59.

Chapter Seventeen Quasi-Periodic Localization on the Zd -lattice

(d > 1) Consider quasi-periodic d-dimensional lattice Schrödinger operators Hλ (x) = λv(x1 + n1 ω1 , . . . , xd + nd ωd )δnn + Δ with Δ the lattice Laplacian on Z , i.e.,

(n ∈ Zd )

(17.1)

d

Δ(n, n ) = 1 if



|nj − nj | = 1

= 0 otherwise We assume v a trigonometric polynomial or real analytic function on Td . More generally, one may consider operators of the form H(x) = λv(x1 + n1 ω1 , . . . , xd + nd ωd )δnn + Sφ

(17.2)

ˆ − n ) is a Toeplitz operator with real analytic symbol. We where Sφ (n, n ) = φ(n always assume ω ∈ Td diophantine. On the Zd -lattice, d > 1, the transfer matrix approach to localization is not available, and in fact, all our results are perturbative. They will be obtained by adaptation of the method developed in Chapters 14 and 15 to the d-dimensional setting. The main result is the following (see [BGS]): Theorem 1. Take d = 2 and assume v real analytic such that, moreover, none of the partial maps v(x1 , ·), v(·, x2 ) is a constant function. Then, for all ε > 0, x ∈ T2 , and λ > λ0 (ε), there is a set of frequencies Fε ⊂ T2 , mes(T2 \Fε ) < ε, such that if ω ∈ Fε , the operator Hλ (x) satisfi es Anderson localization. This result was proven in [BGS] for operators of the form (17.1), and the argument may be extended to case (17.2) as well. One may consider in particular a potential v obtained as v(x1 , . . . , xd ) = w(x1 + x2 + · · · + xd )

(17.3)

with w nonconstant real analytic on T. This situation is considerably simpler because the dynamics happen on T, and localization results were obtained in [CD] for this particular class (d is arbitrary). Thus the next result is not new (the proof sketched below differs from [CD], however). Theorem 2. Consider the Schrödinger operator Hλ = λw(n.ω)δnn + Δ

(17.4)

124

CHAPTER 17

or, more generally, Hλ = λw(nω)δnn + Sφ

(17.5)

on the Z -lattice (d ≥ 1). Here w is a nonconstant real analytic function on T, and φ a real analytic function on Td . Then, for all ε > 0, there is a set of frequencies Fε ⊂ Td s.t. mes(Td \Fε ) < ε, and Hλ satisfi es Anderson localization for ω ∈ Fε , λ > λ0 (ε). Remarks. 1. Operators of the form (17.5) come up naturally in several instances. If we consider the 1D multifrequency Schrödinger operator (on Z) d

H(x) = λφ(x1 + nω1 , . . . , xd + nωd )δnn + Δ the dual operator according to the Aubry duality is indeed H(θ) = 2 cos 2π(θ + nω)δnn + λSφ

(17.6) (17.7)

on the Z -lattice. In particular, if ξ ∈  (Z) is an eigenstate of H(x),   H(x) − E ξ = 0 d

2

then (ξn )n∈Zd given by ζn = e2πinx



ξj e2πij(θ+n.ω)

j∈Z

defines (θ a.s) an extended state of H(θ) with energy E. 2 d Conversely, if ζ ∈  (Z ) is an eigenstate of (17.7), then (ξj )j∈Z given by  ξj = eijθ ζn e2πin(x+jω) n∈Zd

defines (x a.s.) an extended state of H(x). Moreover Spec H = Spec H It follows from Theorem 2 that for λ small enough, (17.7) typically has only point spectrum, and hence (17.6) has only continuous spectrum (this statement, in fact, that only require ω to be diophantine). Thus uses Green’s function estimates on H for diophantine ω in (17.6) and λ sufficiently small, (17.6) has only continuous spectrum. Observe that, in view of Theorem 13.3, this statement (and hence also Theorem 2) is perturbative, and the λ-smallness condition depends on ω. Let E be an eigenvalue of H(θ), and let L(E) denote the Lyapounov exponent of H. Then, necessarily, L(E) = 0. Hence, by continuity of the Lyapounov exponent  = Spec H. Moreover, (see Corollary 7.17), L(E) = 0 on pp H(θ) = Spec H adjustment of the proof of Proposition 12.14 to the multidimensional case permits > 0. Thus mes [L(E) = 0] > 0, implying by us to show that mes (Spec H)  Kotani’s theorem (Proposition 12.5) that ac H(x) = φ for ω ∈ Ωλ ⊂ Td (where mes (Td \Ωλ ) → 0 for λ → 0) and x a.s. 2. As we will explain below, our proof of Theorem 1 in [BGS] does involve some additional difficulties (compared with the D = 1 case), and those were only taken care of for d = 2 at this point.

QUASI-PERIODIC LOCALIZATION ON THE ZD -LATTICE (D > 1)

125

The proof of Theorem 1 does involve the same strategy as in the 1D case. The first result needed deals with Green’s function estimates for fixed energy and at various scales N . Thus, let GN (E, x) = [H[0,N ]2 (x) − E]−1 Our aim is then to obtain the usual bounds GN (E, x) < eN

1−

(17.8)

N (17.9) 10 for all x ∈ ΩN (E) ⊂ T2 with small complement. In the present case, we require estimates of the form c mes [x1 ∈ T (x1 , x2 ) ∈ ΩN (E)] < e−N c mes [x2 ∈ T (x1 , x2 ) ∈ ΩN (E)] < e−N (17.10) 

|GN (E, x)(n, n )| < e−c|n−n | for |n − n | >

for any fixed x1 , x2 ∈ T (which is thus a stronger statement than mes ΩN (E) < c e−N ). This fact will be established following a multiscale procedure according to the method explained in Chapters 14 and 15. There are a number of rather obvious modifications when treating the 2D case. For instance, neighborhoods of sites will now be 2D squares; for technical reason, when exploiting the resolvent identity, one needs, however, to enlarge this class a bit to “fundamental regions.” These include, besides squares, also differences of squares

(see [BGS] for details). A more serious issue and the main difficulty here has to do with the bound #{n ∈ Z2 |n1 | + |n2 | < N and (x1 + n1 ω1 , x2 + n2 ω2 ) ∈ ΩN0 (E)} < N b (17.11) for some b < 1 and log N0 log N . This is a key element in the method (and the only difficulty for generalizing to arbitrary dimension). Ensuring this fact will require us to make a new arithmetic specification of ω, which is more involved than the usual diophantine conditions (also generically–in the measure sense–satisfied). This condition is explained in Lemma 17.12 below. Observe that A = ΩN0 (E) may be considered semialgebraic of degree at most N0C and satisfying, in addition, by (17.10), mes Ax1 < e−N0 and mes Ax2 < e−N0 for all x1 , x2 ∈ T c

c

126

CHAPTER 17

The relevant lemmas are the following: Lemma 17.12. Fix a large positive integer N . There is a subset ΩN ⊂ [0, 1]2 with mes([0, 1]2 \ΩN ) < e−

√ log N

(17.13)

and such that any ω = (ω1 , ω2 ) ∈ ΩN has the following property. Let q1 , q1 , q2 , q2 ∈ Z be bounded in absolute value by N and suppose that the numbers ⎧ θ 1 = q1 ω 1 (mod 1) ⎪ ⎪ ⎪ ⎨θ  = q  ω 1 1 1 (17.14) ⎪ θ = q 2 2 ω2 ⎪ ⎪ ⎩  θ2 = q2 ω2 satisfy |θi |, |θi | < N −1+δ

and −N

with δ small enough. Then

−3+δ

$

θ < 1 θ2

(i = 1, 2)

θ1 < N −3+δ θ2

(17.15)

(17.16)



gcd(q1 , q1 ) > N 1−δ  gcd(q2 , q2 ) > N 1−δ

(17.17)

where δ  (δ) → 0 for δ → 0. Observe that the intuitive meaningof Lemma 17.12   is that ΩN does not  contain too  many see (17.17) small triangles, see (17.15) of very small area see (17.16) . Having introduced ΩN , we may then state Lemma 17.18. Let A ⊂ [0, 1]2 be semialgebraic of degree B . Assume mesAx1 < η, mesAx2 < η for all x1 , x2 ∈ [0, 1]

(17.19)

where Axi denotes the section of A. Let log B log N log

1 η

(17.20)

Let ω ∈ ΩN ⊂ [0, 1]2 be the set introduced in Lemma 17.12. Then, for some b < 1, #{(n1 , n2 ) ∈ Z2 |ni | ≤ N and (n1 ω1 , n2 ω2 ) ∈ A(mod 1)} < N b (17.21) The idea behind Lemma 17.18 is the following: From (17.19), A has to be very close to an algebraic curve Γ, and violation of (17.21) would create too many small and nearly flat triangles–impossible by Lemma 17.12. Once (17.11) is obtained, the Green’s function estimates may be derived by the same arguments used in Chapter 14 in a straightforward way. With the Green’s function estimates (17.8) through (17.10) at hand, proving localization results is again achieved by the argument from Chapter 10 using semialgebraic set theory. The 1D treatment given in Chapter 10 may be adjusted rather easily to the 2D setting (the semialgebraic set analysis again uses the results from Chapter 9 but is slightly more complicated here; see [BGS]).

127

QUASI-PERIODIC LOCALIZATION ON THE ZD -LATTICE (D > 1)

Proof of Lemma 17.12 (sketch). Since by (17.14) and (17.15) |θi | = qi ωi  = |qi ωi − mi | < N −1+ |θi | = qi ωi  = |qi ωi − mi | < N −1+ an initial diophantine restriction of ω permits us to assume that |qi |, |qi | > N 1− Restricting ωi to a size − N12 interval, the number of pairs (q, m), |q| < N , satisfying |qωi − mi | < N −1+ is at most N 0+ . Consider next the condition (17.16). Write ωi = ωi,0 + κi , |κi | < ωi,0 . Thus (17.16) q1 ω1 − m1 q1 ω1 − m1 −3+ < N −3+ −N < q2 ω2 − m2 q2 ω2 − m2

1 N2 ,

and fix

has the form (17.22) |(q1 q2 − q1 q2 )κ1 κ2 + α1 κ1 + α2 κ2 + β| < N −3+   2 Assuming q1 q2 − q1 q2 = 0, (17.22) restricts κ = (κ1 , κ2 ) ∈ 0, N12 to a set of measure at most N −3+ |q1 q2 − q1 q2 |

(17.23)

|q1 q2 − q1 q2 | > N 1+ε

(17.24)

Fix ε > 0, and assume first Then (17.23) < N −4−ε+ . Summing over N 2 × N 2 size N12 -intervals in [0, 1]2 and N 0+ pairs (qi , mi ), (qi , mi ), the total measure contribution in ω-parameter is thus < N −4−ε+ N 4 N 0+ < N −ε+ Next, consider the case |q1 q2 − q1 q2 | < N 1+ε

(17.25)

and assume that min gcd(qi , qi ) < R < N 1−δ



i=1,2

(17.26)

Estimate the measure of ω = (ω1 , ω2 ) for which there are q1 , q1 , q2 , q2 satisfying qi ωi  < N −1+

qi ωi  < N −1+

(17.27)

and (17.25) and (17.26). Let ri = gcd(qi , qi ), qi = ri Qi , qi = ri Qi , (Qi , Qi ) = 1. From (17.25) |Q1 Q2 − Q1 Q2 |
N 1− where

K = {(n1 , n2 ) ∈ Z2 |ni | ≤ N and (n1 ω1 , n2 ω2 ) ∈ A(mod 1)}

Since mes A < η, dist (x, ∂A) < η 1/2 for all x ∈ A. Thus, by the uniformization theorem and (17.20), we may assume that #K1 > N 1− with

(17.32)

  K1 = {(n1 , n2 ) ∈ K dist (n1 ω1 , n2 ω2 ), Γ < η 1/2 }

where “dist” refers to the distance on T2 , and Γ is parametrized by γ : [0, 1] → Γ, with |γ  | < 1, |γ  | < 1 1 Fix δ > 0. The curve Γ may be covered by N 1−δ discs Dα of radius N 1−δ and δ+ each such disc contains (n1 ω1 , n2 ω2 ) (mod 1) for at most N pairs (n1 , n2 ) ∈ K1 . Therefore, since also Var (γ  ) < 1, we may find some α such that the following holds

#Kα > N δ−

(17.33)

129

QUASI-PERIODIC LOCALIZATION ON THE ZD -LATTICE (D > 1)

where

Kα = {n ∈ K1 (n1 ω1 , n2 ω2 ) ∈ Dα }

and if P0 , P1 , P1 ∈ Dα ∩ Γ are distinct points, then    det P1 − P0 , P1 − P0 < N −1+δ |P1 − P0 | |P1 − P0 |

(17.34)

Observe also that from assumption (17.19) and semialgebraicity max #{n1 ∈ Z |n1 | < N and n1 ω1 ∈ Ax2 (mod 1)} < CB < N 0+ x2 max #{n2 ∈ Z |n2 | < N and n2 ω2 ∈ Ax1 (mod 1)} < N 0+ (17.35) x1

2

1 Covering Γ ∩ Dα by at most N δ−δ discs of radius N 1−δ 2 , (17.33) and (17.35) permit us to find n ¯ ∈ Kα , n ∈ Kα s.t. n ¯ 1 = n1 , n ¯ 2 = n2 , and

(n1 − n ¯ 1 )ω1  + (n2 − n ¯ 2 )ω2  < N −1+δ 

Fix n ¯ , n ∈ Kα , and further let n ∈ Kα be a variable site, i = 1, 2 qi = ni − n ¯i

θi = qi ω i

(mod 1)

¯i qi = ni − n

θi = qi ωi

(mod 1)

2

ni

(17.36) = n ¯ i . Define for

and satisfying |θ1 | + |θ2 | < N −1+δ

2

|θ1 | + |θ2 | < N −1+δ

(17.37)

and from (17.34), θ1 θ1 −1+δ (|θ1 | + |θ2 |)(|θ1 | + |θ2 |) < N −3+3δ θ2 θ2 | < N Taking δ small enough, Lemma 17.12 implies (since ω ∈ ΩN ) that 

gcd(qi , qi ) > N 1−δ for i = 1, 2 



< Nδ , Write thus qi = ri Qi , qi = ri Qi with |ri | > N 1−δ ; hence |Qi | + |Qi | ∼   δ   (Qi , Qi ) = 1. Take ki , ki ∈ Z ∩ [0, N ] s.t. ki Qi + ki Qi = 1. It follows that 

1

ri ωi  = (ki Qi +ki Qi )ri ωi  ≤ |ki | qi ωi +|ki | qi ωi  < N δ N −1+δ < N − 2 . Therefore, |θi | = qi ωi  = Qi ri ωi  = |Qi | ri ωi  |θi | = |Qi | ri ωi  so that |qi | |θi | =  |θi | |qi |

130

CHAPTER 17

|θi | =

|qi | |qi |

|θi | ≤ N

|θi | |qi |

2

Recalling (17.37), |θi | < N −1+δ and |qi | > N 1−δ qi ωi 

=

|θi |

< NN

−2(1−δ 2 )+

2

+

(17.38) . Therefore, by (17.38),

< N −1+2δ

2

+

(17.39) 2δ 2 +

qi

But the number of ∈ Z ∩ [0, N ] satisfying (17.39) is at most N r; hence 4δ 2 +    Kα − n ¯ contains at most N elements n − n ¯ = (q1 , q2 ). This contradicts (17.33) and proves Lemma 17.18. .............................................. ............ ......... .... ........ ....... .... ....... ...... ..... ...... ...... ..... ...... ..... ..... ..... .... . ..... . . .... ..... .. .... ... ...... α . . . . . ... ..... ... .... ... ...... ..... ... ...... .. ..... ................... ..... ... ..... ................ ......... ... ..... .... ................ ... ... ..... .. ............... ....... ...... ... ................ . . . . . . . . . . . . . . . . . . . . . . .. . ... ... ............. ... ... ................................... ...... .. ...... .................. .. ..... .. ................... . . . . .... . . .. ..................... ... .. ........................................ 0 .. ... .. ... ... .. ... ... . . ... ... ... . . ... . ... ... ... .. ... ... ... ... ... ... ... .. .... ... . . .... .... ..... ..... ..... ..... ...... ...... ...... ........ ....... .......... ........ ......... ................ ..................................

D

ξ

ξ

ξ

Γ ξ0 = (¯ n1 ω1 , n ¯ 2 ω2 ) ξ = (n1 ω1 , n2 ω2 )

(mod 1)

ξ  = (n1 ω1 , n2 ω2 )

Next, a few comments on the proof of Theorem 2, which is easier (in particular there is no need for Lemma 17.12). Define for θ ∈ T H(θ) = λw(n · ω + θ) + Δ and GN (E, θ) = [H[0,N ]d (θ) − E + i0]−1 Then the bounds(17.8) and (17.9) are obtained for θ ∈ ΩN (E) ⊂ T with c mes T\ΩN (E) < e−N . For log log N log N0 log N , the bound (17.11) becomes (17.40) #{n ∈ [0, N ]d ∩ Zd θ + n.ω ∈ ΩN0 (E)} < N0C < N 0+ Indeed, T\ΩN0 (E) is a union of at most N0C intervals of size < e−N0 and (17.40) only requires a diophantine condition on ω. c

QUASI-PERIODIC LOCALIZATION ON THE ZD -LATTICE (D > 1)

131

References [BGS] J. Bourgain, M. Goldstein, W. Schlag. Anderson localization for Schrödinger operators on Z2 with quasi-periodic potential, Acta Math. 188 (2002), 41–86. [C-Di] V. Chulaevsky, E. Dinaburg. Methods of KAM theory for long-range quasi-periodic operators on Zn . Pure point spectrum, Comm. Math. Physics 153(3) (1993), 559–577.

Chapter Eighteen An Approach to Melnikov’s Theorem on Persistency of Nonresonant Lower Dimension Tori The problem is that of persistency of b-dimensional tori in R2b × R2r -phase space for a real analytic Hamiltonian H of the form H = H(I, θ, y) = H(I1 , . . . , Ib , θ1 , . . . , θb , y1 , . . . , yr ) r  μs |ys |2 + |I|2 + εH1 (I, θ, y) = λ0 , I +

(18.1)

s=1

(the last term is perturbative), as considered in [E], [Kuk], [Pos], and [B1]. Here, I = (I1 , . . . , Ib ), θ = (θ1 , . . . , θb ) are action-angle variables for the “tangential” part of the phase space, and y = (y1 , . . . , yr ) are the “normal” coordinates. The vector λ0 is diophantine and satisfying certain nonresonance conditions; in [E] and [Kuk], the conditions imposed are

λ0 , k − μs = 0

(k ∈ Zb , s = 1, . . . , r)

(18.2)

and also

λ0 , k + μs − μs = 0

(s = s )

(18.3)

In [B1], the persistency result (as stated in [E]) is obtained assuming only (18.2), which is, perturbatively speaking, the natural condition (and does not exclude multiplicities of the normal frequencies–often present in the PDE context). We will treat here only the problem of a Hamiltonian perturbation of a linear equation, to which (18.1) may be reduced. Consider thus the equation 1 ∂H q˙j = (1 ≤ j ≤ B) (18.4) i ∂ q¯j with Hamiltonian H(q, q¯) =

B 

μj |qj |2 + εH1 (q, q¯)

(18.5)

j=1

with canonical coordinates q1 , . . . , qB ∈ C. {μj } are real, and we assume for simplicity that H1 (q, q¯) is given by a polynomial in qj , q¯j with real coefficients. Let 1 ≤ b < B, and consider λ = (μ1 , . . . , μb ) as a b-dim parameter (in the context (18.1), these parameters are extracted from the |I|2 -term by amplitudefrequency modulation; see [B3]). For simplicity, we assume that the remaining frequencies μj (b < j ≤ B) are constant.

134

CHAPTER 18

Letting ε = 0, (18.4) has the (unperturbed) quasi-periodic solution qj (t) = aj eiλj t =0

(1 ≤ j ≤ b) (b < j ≤ B)

(18.6)

There is the following persistency result: Theorem 18.7. L et λ be restricted to an interval Ω ∈ Rb , and assume that ε in (18.5) is sufficiently small. Fix a1 , . . . , ab ∈ R+ . There is a Cantor subset ε→0 Ωε ⊂ Ω, mes (Ω\Ωε ) → 0, and a smooth map λ → λ on Ω, s.t. for λ ∈ Ωε , and there is a quasi-periodic solution of (18.4)   qj (t) = qˆj (k)eik.λ t (1 ≤ j ≤ B) (18.8) k∈Zb

satisfying qˆj (ej ) = aj

(1 ≤ j ≤ b)

|ˆ qj (k)| < e−c|k| for some c > 0 and



|ˆ qj (k)|
0 This constant will decrease slightly along the iteration but remain bounded away from 0. With qr defined as a C 1 -function on the entire (λ, λ )-parameter space, application of the implicit function theorem to the Q-equations ∂H1 ε  λj = λj + (ej ) with q = qr aj ∂ q¯j yields λ = λ + εϕr (λ) with ∂ϕr  < C which graph we denote by Γr . Clearly, by (ii), < εq |ϕr − ϕr−1 | ∼ (r) − q(r−1)  < εδr so that Γr is an εδr -approximation of Γr−1 .

137

AN APPROACH TO MELNIKOV’S THEOREM ON PERSISTENCY

(iv) There is a collection Λr of intervals I in R2b of size A−r s.t. C

(a) On I ∈ Λr , qr (λ, λ ) is given by a rational function in (λ, λ ) of degree at 3 most Ar . % (b) For (λ, λ ) ∈ I∈Λr I, F (qr ) < κr ∂F (qr ) < κ ¯r

(∂ refers to λ- or λ -derivatives)

1 Remark. Again, log log κr +¯ κr ∼ r. % (c) For (λ, λ ) ∈ I∈Λr I, T = Tq(r−1) satisfies

TN−1  < Ar

C 

|TN−1 (k, k  )| < e−c|k−k | for |k − k  | > rC where TN refers to the restriction |k| < N = Ar . Remark. Since ε is small, the control of TN−1 at initial scales reduces to the diagonal D and hence minoration of the expressions |k.λ ± μj |, |k| ≤ N . (d) Each I ∈ Λr is contained in an interval I  ∈ Λr−1 and     mes b Γr ∩ ( I \ I) < A−r/5 I  ∈Λr−1

(18.19)

I∈Λr

... ...... ....... ....... ...... ...... ....... . . . . . . .. ....... ....... ...... ...... ....... ...... ...... ....... . . . . . . .. ........ ....... ........ ........ ........ ....... ...... ....... . . . . . . ... ......... ....................................... ....................................

Γr

I I

III. Invertibility of Tq(r) Let N = Ar+1 , which may be assumed to be large based on the by previous remark. To construct Δr+1 q, we need to control (RN Tq(r) RN )−1 with a further restriction of the%(λ, λ )-parameter set. Fix (λ, λ ) ∈ I∈Λr I. Since 1 ∼r δr that RAr Tq(r) RAr also satisfies

< Δ q < δ , log log Tq(r) − Tq(r−1)  ∼ r r

it follows from the assumption on RAr Tq(r−1) RAr the estimate in (c). Fix M0 , satisfying M0 ∼ (log N )C/2

(18.20)

138

CHAPTER 18

(for some C to be specified). Assume that the following estimates are obtained for any interval J ∈ Zb of size M0 and centered at some k ∈ Zb , 21 Ar < |k| < Ar+1 1−

(RJ Tq(r) RJ )−1  < eM0

(18.21)



|(RJ Tq(r) RJ )−1 (k, k  )| < e−c|k−k | for k, k  ∈ J, |k − k  | >

M0 10

(18.21 )

It will then follow from the resolvent identity that 1−

(RN Tq(r) RN )−1  < Ar + eM0 C

(18.22)

and 

|(RN Tq(r) RN )−1 (k, k  )| < e−c|k−k | for |k − k  | > rC

(18.22 )

To obtain (18.21) and (18.21), we may perturb q (r) by 0(e−M0 ). Thus we may replace q (r) by q (r0 ) with r0 < r satisfying δr0 < e−M0

(hence r0 ∼ log M0 )

(18.23)

Consider the following matrix: T σ = Dσ + Sq(r0 ) (λ,λ )

(18.24)

where we introduced an additional (1-dim) parameter σ and σ D±,j,k = ±(k.λ + σ) − μj

(18.25)



Hence (18.21) and (18.21 ) will follow from −1

σ −1 (TM )  < eM0 0

(18.26)

and 

σ −1 ) (k, k  )| < e−c|k−k | for |k − k  | > |(TM 0

M0 10

(18.26 )

valid for all σ ∈ {k.λ | 12 Ar < |k| < Ar+1 }. Lemma 18.27. Assume λ diophantine. Then (18.26) and (18.26 ) hold for all σ 1/2 outside a set of measure at most e−M0 (and depending on λ, λ ). Remark. It is straightforward to ensure the required diophantine properties on λ  by introducing the interval systems Λr appropriately. Proof of Lemma 18.27. Follow the multiscale argument from Chapter 14, complexifying in the parameter σ. The details are straightforward. (Observe that these operators are closely related to those considered in Chapter 17, Theorem 2). Now fix I ∈ Λr0 , and consider the set σ S = {(λ, λ , σ) ∈ I × R (18.26) + (18.26) fail for TM } 0

(18.28)

This condition may be expressed by a polynomial in the matrix elements of T σ of degree < M0C . These matrix elements are at most linear in σ and rational 3 functions in λ, λ of degree at most Ar0 (by assumption (a) on q (r0 ) ). Therefore, S 3 is semialgebraic of degree at most M0C Ar0 .

139

AN APPROACH TO MELNIKOV’S THEOREM ON PERSISTENCY r03



Since Γr0 ∩ I is defined by an equation in λ, λ of degree < CA , S ∩ (Γr0 ∩ 3 I) × R) also is semialgebraic of degree < M0C Ar0 . By Lemma 18.27, each (λ, λ )1/2 section is of 1-dim measure < e−M0 so that    1/2 (18.29) mes 1+b S ∩ (Γr0 ∩ I) × R) < e−M0 Our aim is to estimate for |k| > 12 Ar    mes b {λ| λ, λ + εϕr0 (λ), k. λ + εϕr0 (λ)) ∈ S ∩ (I × R)}   Apply first decomposition Lemma 9.9 to the set S ∩ (Γr0 ∩ I) × R as subset of (Γr0 ∩ I) × R (which may be identified with an interval in Rb+1 ). This gives a subset Γ ⊂ Γr0 ∩ I of measure   3 mes b Γ < (M0 Ar0 )C A−r < A−r/2 see (18.20) and (18.23) (18.30) such that for all k, |k| > 12 Ar     mes b {λ (λ, λ+εϕr0 (λ), k. λ + εϕr0 (λ) ∈ S ∩ (Γr0 \Γ ) ∩ I) × R } (18.31) 1/3

< e−M0 Therefore, there is a set Γ ⊂ Γr0 ∩ I

1/3

mes b Γ < A−r/2 + Ab(r+1) e−M0

< A−r/3

(by choice (18.20) of M0 ) such that (18.26) and (18.26) hold for all (λ, λ ) ∈ (Γr0 \Γ ) ∩ I and σ = k.λ , 12 Ar < |k| < Ar+1 . From the previous discussion, this implies (18.21) and (18.21) on (Γr0 \Γ ) ∩ I. Letting I range over Λr0 , the total measure removed from Γr0 is at most r0C A .A−r/3 < A−r/4 . Since (18.21) and (18.21) allow 0(e−M0 ) perturbation of (λ, λ ) and Γr0 , Γr are at distance < δr0 < e−M0 , by (18.23), we obtain a subset r ⊂ Γr , mes b Γ r < A−r/4 such that (18.21) and (18.21) hold on Γ    r ) I ∩ (Γr \Γ (18.32) I∈Λr0

and hence on

 

 r ) I ∩ (Γr \Γ

(18.33)

I∈Λr

Since on (18.33), as well as (18.22) and 18.22 ), holds with N replaced by Ar , (18.22) and (18.22) hold on (18.33). Clearly, since the bound in (18.22) is at most C r+1 C/2 C C Ar +e(log A ) < A(r+1) , (18.22) and (18.22) remain valid on an A−(r+1) neighborhood of (18.33). This gives a collection Λ r+1 of intervals in R2b of size C A−(r+1) , s.t. for (λ, λ ) ∈ I ∈ Λr+1 (RAr+1 Tq(r) RAr+1 )−1  < A(r+1)

C

(18.34)



and (18.22 ), and   mes b (I ∩ Γr )\ I∈Λr

 I  ∈Λr+1

 r < A−r/4 (I  ∩ Γr ) ≤ mes b Γ

(18.35)

140

CHAPTER 18

which will imply (18.19) at stage r + 1. IV. Construction of q (r+1) % Denoting T = Tq(r) , N = Ar+1 , then define for (λ, λ ) ∈ I∈Λr+1 I   Δr+1 q = −TN−1 F (qr ) Thus Δr+1 q is a rational function in (λ, λ ) of degree at most 3

3

N C .Ar + CAr < Ar

3

+Cr

< A(r+1)

3

We have from (18.34) and assumption (b) C

Δr+1 q < A(r+1) .κr = δr+1

(18.36)

and ∂(Δr+1 q)

≤ ∂TN−1.F (qr ) + TN−1 ∂F (qr ) < TN−1 2 qr C 1 κr + TN−1 .¯ κr 2(r+1)C

0 taken small enough.) The problem we consider is that of persistency of the unperturbed solution $ qnj (t) = aj eiλj t (1 ≤ j ≤ b) (19.3) qn (t) = 0 for n ∈ {n1 , . . . , nb } of the linear equation (ε = 0). There is the following analogue of the theorem proven in Chapter 18. Theorem 19.4. Consider (19.1) in the setting described above (with arbitrary dimension d). Fix a1 , . . . , ab ∈ R+ . There is a Cantor set Ωε ⊂ Ω, mes (Ω\Ωε ) → 0 for ε → 0 and a smooth map λ → λ on Ω s.t. for λ ∈ Ωε , and there is a quasi-periodic solution of (19.1)   qn (t) = qn (k)eik.λ t (n ∈ Zd ) k∈Zb

satisfying q nj (ej ) = aj

and

< e | qn (k)| ∼

 (n,k)∈R

(1 ≤ j ≤ b) −c(|n|+|k|)

| qn (k)|
rC

(19.6 )

Here T = Tq(r) and TN refers to the restriction of T in both indices k and n, thus TN = R |k| M0−C for k ∈ Zb , 0 < |k| < M0 σ Consider TM with minn∈Q |n| > M0 . There is a system of Lipschitz functions 0 ,Q C1  σs = σs (λ, λ ), s < e(log M0 ) (depending on Q) such that

σs Lip ≤ CM0

(19.14)

and for λ, λ ∈ I 1−

σ (TM )−1  < eM0 0 ,Q

(19.15)

and 

σ |(TM )−1 (ξ, ξ  )| < e−c|ξ−ξ | for |ξ − ξ  | > 0 ,Q

M0 10

(19.15 )

provided c2

min |σ − σs (λ, λ )| > e−M0 s

(C1 , c2 are some constants to which we will refer later on).

(19.16)

146

CHAPTER 19

Lemma 19.13 is proven multiscale in M0 and is mainly based on Lemma 19.10, first-order eigenvalue variation, and the resolvent identity. Assume Lemma 19.13 valid at scale M0 , and let 1c 2 2

M1 = eM0

(19.17)

and Q ⊂ Zb an M1 -interval s.t. minn∈Q |n| > M1 . σ+k.λ Let Q0 = [−M0 , M0 ]d , and consider the matrices TM where |k| < 0 ,n0 +Qo M 1 , n0 ∈ Q Q

n0

..... ..... ..... ..... ..... ..... ..... ..... ..... ... . ... ... . ... . ... . ... . ... . ... . ... . .

k0

M1

Choose B = M1ρ (0 < ρ < 1 to be specified) in Lemma 19.10 and let {πα } be the corresponding partition of Q1 . σ Call a site (±, n, k) singular if |D±,n,k | < 1. Assume (±, n, k) and (±, n , k  )   singular with n ∈ πα , n ∈ πα , α = α . It then follows from (19.9) and (19.12) that |n − n | + |k − k  | > cM1ρ

(19.18)

Therefore, if we denote S = {(n, k) ∈ Q × [−M1 , M1 ]b | (+, n, k) or (−, n, k) is singular} the following separation property holds   > M ρ for α = α dist S ∩ (Zb × πα ), S ∩ (Zb × πα ) ∼ 1

(19.19)

(19.20)



σ+k.λ Fix n0 ∈ Zd . From the induction hypothesis, TM will satisfy (19.15), 0 ,n0 +Q0  (19.15 ) unless c2

min |σ + k.λ − σs (λ, λ )| < e−M0 s

Since k.λ  > M1−C for k ∈ Zb , 0 < |k| < M1 , it follows from (19.17) that, c2 fixed s, |σ + k.λ − σs (λ, λ )| < e−M0 holds for at most one value of k. Thus, 

for for

fixed α, there are clearly at most (recalling (19.11)) M1ρC0 .d × e(log M0 )

1

C1

< M1100

(19.21)



k.λ +σ sites (k, n) ∈ [−M1 , M1 ]b ×πα for which (TM )−1 fails (19.15) and (19.15). 0 ,n+Q0   b For these (n, k), necessarily (k + [−M0 , M0 ] ) × (n + Q0) ∩ S = φ. Partitioning Ω = [−M1 , M1 ]b × Q in intervals of size M0 , the preceding permits us to perform a decomposition

Ω = Ω 0 ∪ Ω1

(19.22)

APPLICATION TO THE CONSTRUCTION OF QUASI-PERIODIC SOLUTIONS

with Ω1 =



Ω1,β

147

(19.23)

β

where Ω0 , Ω1β are unions of M0 -interval 1

diam Ω1,β < M1100 dist (Ω1,β , Ω1,β  ) >

+ρ

+ M1ρC0

M1ρ

for β = β

(19.24) 

(19.25)

and (TΩσ0 )−1 satisfies (as a consequence of the resolvent identity) −1

(TΩσ0 )−1  < eM0

(19.26)

and 

|(TΩσ0 )−1 (ξ, ξ  )| < e−c|ξ−ξ | for |ξ − ξ  | > 10M0

(19.26 )

Notice here that clearly (19.26) and (19.26) allow an e−M0 -perturbation of T σ and hence an 0(M1−1 e−M0 )-perturbation of (λ, λ , σ). The decomposition (19.22) and (19.23) thus may be used on an M1−1 e−M0 -neighborhood of an initial parameter 3 1,β an M 4 ρ -neighborhood of Ω1,β . If we ensure that choice (λ, λ , σ). Denote Ω 1

(TΩσ )−1  1,β

ρ/2

< eM1

(19.27)

for all β, (19.22), (19.23), (19.25), (19.26), (19.26 ), (19.27), and the resolvent identity imply that ρ/2

(TΩσ )−1  < e3M1

1−

< eM1

(19.28)

and 

|(TΩσ )−1 (ξ, ξ  )| < e−c|ξ−ξ | for |ξ − ξ  | >

M 10

(19.28 )

Consider condition (19.27). Recall that for (k, n) ∈ Ω, σ = ±(k.λ + σ) − |n|2 = ±σ − |n|2 + 0(M1 ) D±,n,k

with n ∈ Q; hence |n| > M1 . Assume σ > 0, thus | − σ − |n|2 | > M12 . Hence (R− TΩσ

1,β

and (TΩσ

1,β

< M −2 R− )−1  ∼ 1

)−1 is controlled by the inverse of the self-adjoint matrix

R+ TΩσ

1,β

R+ − ε2 R+ Sq R− (R− TΩσ

1,β

R− )−1 R− Sq R+

= σ + R+ TΩ1,β R+ − ε2 R+ Sq R− (R− TΩσ

1,β

= σ + T  (λ, λ , σ)

(19.29)

R− )−1 R− Sq R+ (19.30)

Clearly, ∂λ T   ≤ ∂λ q < C, and similarly, ∂λ T   < M1 . Also, ∂σ T   < Cε2 (R− TΩσ R− )−1 2 < M1−4 . 1,β

Let {Ei (T  )} be the ascending ordering of the eigenvalues of T  . Then Ei are continuous functions of the parameters λ, λ , σ and (from the analyticity assumption

148

CHAPTER 19

on q = q(λ, λ )) piecewise holomorphic in each (1-dim) parameter component separately. From the preceding and first-order eigenvalue variation, it follows that Ei (λ, λ , σ) is Lipschitz and Ei Lip(λ) < C, Ei Lip(λ ) < M1 , Ei Lip(σ) < M1−4

(19.31)

From (19.30), {σ + Ei (λ, λ , σ)} is a parametrization of Spec (19.29). Fix i. By (19.31), the equation σ + Ei (λ, λ , σ) = 0 defines a function σ = σi (λ, λ ) Thus

  σi (λ, λ ) + Ei λ, λ , σi (λ, λ ) = 0

and again by (19.31), ¯ λ ¯  )| ≤ M −4 |σi (λ, λ ) − σi (λ, ¯ λ ¯  )| + C|λ − λ| ¯ + M1 |λ − λ ¯| |σi (λ, λ ) − σi (λ, 1

implying σi Lip(λ) ≤ C, σi Lip(λ ) < M1

(19.32)

Moreover,

  |σ + Ei (λ, λ , σ)| = |σ − σi (λ, λ ) + Ei (λ, λ , σ) − Ei λ, λ , σi (λ, λ ) |   = |σ − σi (λ, λ )| 1 + 0(M1−4 )

Consequently,

  dist Spec (19.29), 0 ∼ min |σ − σi (λ, λ )| i

Hence )−1  ≤ max |σ − σi (λ, λ )|−1

(TΩσ

1,β

i

(19.33)

Collecting the functions {σi } over all β, we thus obtain at most M1b+d Lipschitz functions {σi = σi (λ, λ )} such that < M σi Lip ∼ 1

(19.34)

and max (TΩσ β

1,β

)−1  ≤ max |σ − σi (λ, λ )|−1 i

ρ/2

(19.35)

If (19.35) < eM1 , (19.27) will hold, hence (19.28) and (19.28). The preceding construction of the {σi}-functions has been carried out after restriction of the (λ, λ , σ)-parameters to an M1−1 e−M0 -neighborhood of an initial choice. As pointed out earlier, one may indeed keep the same decomposition (19.22) and (19.23) on such a neighborhood. The number of these parameter neighborhoods clearly may be bounded by M1d+1 (M1 eM0 )2b+1

APPLICATION TO THE CONSTRUCTION OF QUASI-PERIODIC SOLUTIONS

149

and the total number of {σi } functions therefore is at most M1b+d M1d+1 (M1 eM0 )2b+1 < e2(b+1)M0 < e2(b+1)(log M1 )

2/c2

by (19.17). From the preceding, if ρ/2

min |σ − σi (λ, λ )| < e−M1 i

then (19.28) and (19.28) hold. 1 C1 Recalling condition (19.21) on ρ, i.e., M1ρC0 d eC(log log M1 ) < M1100 , we take ρ=

1 200C0 .d

(19.36)

This proves Lemma 19.19 with c2 = ρ2 , C1 = 3/c2 . σ Next, we will prove an estimate on (TM )−1 with Q = [−10M0, 10M0 ]d . To 0 ,Q establish this estimate, some further restrictions in (λ, λ )-parameter space will be needed. In view of this, we make the following definition: Definition. Say that A ⊂ [0, 1]2b has sectional measure at most ε, messec A < ε, if the following holds: Let ϕ : [0, 1]b → [0, 1]b be C 1 with ∇ϕ < 10−2 . Then   mes [λ ∈ [0, 1]b | λ, λ + ϕ(λ) ∈ A] < ε (19.37) As part of a multiscale reasoning, we next prove Lemma 19.38. Let T = Tq with q = q(λ, λ ) be holomorphic in λ, λ on an interval I ⊂ [0, 1]2b and ∂q < C . Let M0 ∈ Z+ be a large integer 1

M0 < M1 < exp exp(log M0 ) 10

(19.39)

Assume λ satisfi es a DC k.λ  > M1−C for |k| < M1 σ σ  Assume that TM = TM,Q=[−10M,10M] d satisfi es for M < M0 and (λ, λ ) ∈ I the following property σ −1 (TM )  < eM

1−

(19.40)

and 

σ −1 |(TM ) (ξ, ξ  )| < e−c|ξ−ξ | for |ξ − ξ  | >

1 M 10

(19.40 )

for all σ outside a set of measure < e−M

c3

(19.41)

Then there is a subset A of I obtained as a union of intervals of size [exp exp(log log M1 )3 ]−1 , such that messec A < [exp exp(log log M1 )2 ]−1

(19.42)

150

CHAPTER 19

and if (λ, λ ) ∈ I\A 1 10

σ −1 (TM )  < eM1 1

(19.43)

for σ outside a set (depending on λ, λ ) of measure c4

< e−M1

(c4 = 10−7 d−1 c2 )

(19.44)

Proof. Let M = exp(log log M1 )3 < exp(log M0 )3/10

(19.45)

Start by paving [−M1 , M1 ] × [−10M1, 10M1 ] by translates of [−M, M ]b × σ+k.λ −1 [−M, M ]d . Thus we need to control (TM,Q ) with |k| < M1 and Q ⊂ d [−10M1 , 10M1] an M -interval. If minn∈Q |n| > M , use Lemma 19.13 applied σ+k.λ −1 at scale M . For (TM,Q ) to satisfy (19.15) and (19.15), we need thus to satisfy b

d

min |σ + k.λ − σs (λ, λ )| > e−M

c2

(19.46)

s

where {σs } is the system of Lipschitz functions introduced in Lemma 19.13. These depend on Q. Collecting then over all M -intervals Q in [−10M 1 , 10M1]d \[|n| < M ], their number is at most (20M1 )d .e(log M)

C1

< M1d+1

(19.47)

and we still denote {σs } this collected system. We introduce A ⊂ I s.t. for (λ, λ ) ∈ I\A and any k ∈ Zb , M 1+ < |k| ≤ M1 , min |kλ − σs (λ, λ ) + σs (λ, λ )| > 2e−M

c2

s,s

(19.48) 

σ+k.λ −1 If (19.48) holds, it will follow that if minn∈Q∪Q |n| > M and (TM,Q ) , 

σ+k ,λ −1 ) both fail (19.15) and (19.15), then |k − k  | < M 1+ . Thus, if (TM,Q  

σ+k.λ −1 ) fails (19.15) and (19.15), then k is within minn∈Q |n| > M and (TM,Q an M 1+ -neighborhood of a single ¯k ∈ [M1 , M1 ]b . Since necessarily

min

|k −k| 0, s.t. |k − ¯k | + min |n|2 − R2 < M 1+

(19.49)

n∈Q



σ+k.λ −1 Unless (19.49) holds, (TM,Q ) will satisfy (19.15) and (19.15). 2(d+1)

= M1b+2d+2 Returning to (19.48), there are by (19.47) at most M 1b .M1 conditions involved. We claim that if |k| > M 1+ and ϕ : [0, 1]b → [0, 1]b satisfies |∇ϕ| < 10−2 , then      c2 c2 mes [λ |k. λ+ϕ(λ) −σs λ, λ+ϕ(λ) +σs (λ, λ+ϕ(λ) | ≤ 2e−M ] < e−M (19.50)

APPLICATION TO THE CONSTRUCTION OF QUASI-PERIODIC SOLUTIONS

151

This will imply that mes sec A < M1b+2d+2 e−M

c2

< [exp exp(log log M1 )2 ]−1

(19.51)

Clearly, A may moreover be taken to be a union of intervals of size e −M .  1+ Tocheck (19.50), assume, for   instance, that k1 > M . Considering k. λ + ϕ(λ) −σs λ, λ+ ϕ(λ) + σs λ, λ+ ϕ(λ) as a function in λ1 with other variables fixed, we have   99 ∂λ1 [k. λ + ϕ(λ) ] > k1 > M 1+ 100 while by (19.14)   < M (σs − σs ) λ, λ + ϕ(λ) Lip(λ1 ) ∼ This clearly implies (19.50). σ+k.λ , |k| < M1 . Use the induction hypothesis and in particular Consider next TM 2 (19.41) at scale M 2 < M0 . Since conditions (19.40) and (19.40) are semialgebraic in σ of degree < (CM )b+d and λ assumed diophantine as specified in Lemma σ+k.λ −1  C 19.38,  it follows that (TM 2 ) fails (19.40) and (19.40 ) for at most M values of k we use here (19.45) . Partitioning Ω = [−M1 , M1 ]b × [−10M1, 10M1 ]d in M -intervals, the preceding and the resolvent-identity provide a decomposition Ω = Ω 0 ∪ Ω1

(19.52)

where Ω0 , Ω1 are unions of M -intervals #π1 (Ω1 ) < M C + M 1+

(19.53)

(π1 = projection in k-variable) and M 1+ (k, n) ∈ Ω1 ⇒ |n| < M 2 or |n| − R < R

(19.54)

and such that (TΩσ0 )−1  < eM

2

(19.55)

M1

R

M2

..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ... . ... ... . ... . ... . ... . ... . ... . ... . ... . .

¯k

M1

σ Our purpose is to apply Proposition 14.1 with A(σ) = T M . Thus B1 < C. Take 1 in (14.5) Λ = Ω1 satisfying by (19.53) and (19.54)

#Λ < M C .Rd−2+

(19.56)

152

CHAPTER 19 2

and B2 = eM by (19.55). To obtain condition (14.6), simply apply Lemma 19.13, the induction hypothesis at scale M0 < M1 , together with the resolvent identity. Thus c3 C1 c2 1 σ −1 mes [σ (TM )  > B3 = eM0 ] < M1b+d e(log M0 ) e−Mo + M1b .e−M0 1 B2 (19.57) We make the following distinction (not necessary for d ≤ 2) 1

Case I: R ≤ M1100d 1

Case II: R > M1100d 1

Case I. By (19.56), #Λ < M1100 . Proposition 14.1 then permits us to conclude that   1 1 cM110 σ −1 M110 mes [σ (TM )  > e ] < exp − (#Λ) log(M1 +B2 +B3 ) 1 (19.58) 1 − 1 10 100

< e−cM1

M0−1

1 15

< e−M1

Case II. Then apply Proposition 14.1 to A(σ) = T σ

−2 d−1

M1 ,[|n|M110

define

]

)−1 , we simply use Lemma 19.13. More precisely, M2 = M110

−4 −1

d

(19.60)

−3 −1 [M110 d

and cover [|k| < M1 ] × < |n| < 10M1 ] by (b + d)-dim intervals P of size M2 . From Lemma 19.13, applied at scale M2 , 1−

(TPσ )−1  < eM2

(19.61)



|(TPσ )−1 (ξ, ξ  )| < e−c|ξ−ξ | for ξ, ξ  ∈ P, |ξ − ξ  | >

M2 10

(19.61 )

for σ outside a set of measure at most c2 10−5 d−1

c2

M1b+d .e(log M2 ) .e−M2 < e−M1 C1

(19.62)

From (19.59), (19.60), (19.61), (19.61), and another application of the resolvent identity, we conclude that for σ outside a set of measure at most 10−7 d−1

e−M1

c2 10−5 d−1

+ e−M1

c2 10−7 d−1

< e−M1

(19.63)

one has that 10−4 d−1

σ −1 )  < eM1 (TM 1

1 10

< eM1

(19.64)

153

APPLICATION TO THE CONSTRUCTION OF QUASI-PERIODIC SOLUTIONS

Together with (19.58), this completes the proof. Lemma 19.65. In the statement of Lemma 19.38 we have in addition to (19.43) also the off-diagonal decay estimate  1 σ −1 M1 |(TM ) (ξ, ξ  )| < e−c|ξ−ξ | for |ξ − ξ  | > (19.66) 1 10 c5 and σ outside a set of measure < e−M1 , c5 = c2 10−12 d−1 . −4 −1 Proof. Define again M2 = M110 d . Consider first M2 intervals Q ⊂ [−10M1, 10M1 ]d s.t. min |n| > M2 . Lemma 19.13 implies that n∈Q

1−



σ+k.λ −1 (TM )  < eM2 2 ,Q

(19.67)

and 1 M2 10 for all k ∈ [−M1 , M1 ]b and σ outside a set of measure at most 



σ+k.λ −1 ) (ξ, ξ  )| < e−c|ξ−ξ | for |ξ − ξ  | > |(TM 2 ,Q

c2

10−5 d−1 c2

c2

1

M1b+d .e(log M2 ) .e−M2 < e− 2 M2 < e−M1 C1

(19.67 )

(19.68)

It remains to consider the lower region [−M1 , M1 ] × [−100M2, 100M2 ] . Take M as in the proof of Lemma 19.38. Consider a paving of [−M 1 , M1 ]b × [−100M2, 100M2 ]d with intervals P = [k − M, k + M ]b × [−10M, 10M ]d and intervals P  = [k − M, k + M ]b × Q with Q an M -interval in [−100M2, 100M2]d s.t. minn∈Q |n| > M . b

d

10M1

100M2

P

10M

P M1

For the P -intervals, the assumption in Lemma 19.38 implies again that (T Pσ )−1 will satisfy (19.40) and (19.40) except for M C -many values of k. For the P  intervals, Lemma 19.13 applies again, and the number of “bad” intervals there is at most e(log M)

C1

(100M2)d < M12.10

−4

Thus the total number of “bad” P and P  intervals in [−M1 , M1 ]b × −4 [−100M2, 100M2 ]d is at most M12.10 (for a fixed σ). Define M3 = M110

−3

It is clear from the preceding that for any k ∈ [−M1 , M1 ]b , there is some ¯k ∈ [−M1 , M1 ]b and some M3 < M  < 2M3 s.t. k ∈ ¯k + [−M  , M  ]b ⊂ [−M1 , M1 ]b

154

CHAPTER 19

and

  1/2 1/2 ¯k + ([−M  , M  ]b \[−M  + M3 , M  − M3 ]b )   % % π1 (P ) ∪ π1 (P  ) = φ ∩ P

P bad

(19.69)

bad

Lemma 19.38 permits us to ensure for (λ, λ ) ∈ I\A that 

σ+¯ k.λ −1 )  < e(M (TM 

1  10 )

1 9

< eM3 for all ¯k ∈ [−M1 , M1 ]b and M3 < M  < 2M3 (19.70) except for σ in a set of measure 10−11 d−1 c2

c4

< M3 M1b e−M3 < e−M1 Thus, excluding a σ-set of measure 10−5 d−1 c2

10−11 d−1 c2

(19.71) c5

+ e−M1 < e−M1 (19.72) e−M1 (recalling also (19.68)), properties (19.67), (19.67), (19.69), and (19.70), together with the resolvent identity, imply (19.66). This proves Lemma 19.65. Remarks. 1. Lemma 19.65 permits us to recover at scale M1 the assumptions form Lemma 19.38 at scale M ≤ M0 , with c3 = c5 = 10−12 d−1 c2 . There is, however, an additional restriction (λ, λ ) ∈ I\A, where mes sec A < [exp exp(log log M1 )2 ]−1 2. A comment on the assumption for q = q(λ, λ ) to be holomorphic in λ, λ on I ⊂ [0, 1]2b . Recall that q = q (r) is obtained along the Newton iteration scheme and satisfies for r0 < r 1 ∼ r0 (19.73) q (r) − q (r0 )  < δr0 where log log δr 0 Clearly, one may replace q by q (r0 ) provided δr0 < e−M1 hence r0 ∼ log M1 (19.74) This approximative solution q (r0 ) = q (r0 ) (λ, % λ ) is given by a rational function with bounded derivatives in (λ, λ ) for (λ, λ ) ∈ I∈Λr I, consisting of intervals I of 0

size A−r0 in [0, 1]2b . If Lemmas 19.38 and 19.65 are applied with I one of those intervals, the set A removed from I thus satisfies C (19.75) mes sec A < [exp exp(log r0 )2 ]−1 A−r0 We return now to Chapter 18 and the inductive construction. In order to be able to estimate TN−1 , N = Ar , we add the following hypothesis: C

σ , with (†) Let exp(log log M )3 ≤ r1 ≤ r. Then for all (λ, λ ) ∈ I ∈ Λr1 , TM,Q d Q = [−10M, 10M ] , satisfies σ (TM,Q )−1  < eM

1−

(19.76)

and 

σ |(TM,Q )−1 (ξ, ξ  )| < e−c|ξ−ξ | for |ξ − ξ  | >

for σ outside a set of measure < e−M . c3

M 10

(19.76 )

155

APPLICATION TO THE CONSTRUCTION OF QUASI-PERIODIC SOLUTIONS

We first show how (†) is established inductively at scale r < exp(log log M1 )3 ≤ r + 1

(19.77)

M0 = exp(log log M1 )10

(19.78)

According to (19.39), let If we choose r0 ∼ log M1 , clearly exp(log log M0 )3 < r0 < r σ , Q0 = [−10M0 , 10M0]d , for all (λ, λ ) ∈ and (†) holds thus for TM 0 ,Q0

% I0 ∈Λr0

I0 .

according to Remark (2) We use Lemmas 19.38 and 19.65, replacing q by q above and taking (λ, λ ) ∈ I0 for a fixed I0 ∈ Λr0 . Statements (19.76) and (19.76) with M = M1 then follow from (19.43) and (19.66), provided a subset A = A(I0 ) is removed from I0 , satisfying (19.75) (r0 )

mes sec A(I0 ) < [exp exp(log r0 )2 ]−1

(19.79)

and A composed of intervals of size [exp exp(log log M1 )3 ]−1 ∼ e−r

(19.80)

by (19.77). Consider the curve Γr defined by λ = λ + εϕr (λ) and   (Γr ∩ I) ⊂ (Γr ∩ I0 ) I0 ∈Λr0

I∈Λr

 It follows from (19.79) and % (19.80) that there is Λr ⊂ Λr such that (†) holds for  M = M1 and (λ, λ ) ∈ I∈Λ I, with r     C mes b (Γr ∩ I)\ (Γr ∩ I) < Ar0 [exp exp(log r0 )2 ]−1 I∈Λr

I∈Λr

1 M1 < [exp exp(log r)1/3 ]−1




M 10

(19.82 )

Thus J = ([−M, M ]b + k) × Q, where Q is a size-M interval in [−N, N ]d . N Since TA−1 r already satisfies the required properties, it suffices to consider 2A = 1 r A ≤ |k| ≤ N . 2

156

CHAPTER 19

We have thus 

σ=k.λ RJ T RJ = TM,Q

and we distinguish the cases minn∈Q |n| > M and Q = [−10M, 10M ]d = Q0 . If minn∈Q |n| > M , invoke Lemma 19.13. Condition (19.16) becomes min |k.λ − σs (λ, λ )| > e−M

c2

s

Define r = exp(log log M )3 < r

(19.83)

Restricting (λ, λ ) to the graph Γr , we need to exclude the set of λ’s for which     c2 (19.84) min min |k. λ + εϕr (λ) − σs λ, λ + εϕr (λ) | ≤ e−M s

Q

Since σs Lip ≤ CM and |k|  M , the measure of this set is at most N b+d e(log M) e−M C

c2

1

< e− 2 M

c2

< A−r

provided c2 C6 > 1. For Q = Q0 = [−10M, 10M ]d, we argue again as in Chapter 18, considering the semialgebraic sets σ S = {(λ, λ , σ) ∈ I  × R|TM,Q fails (19.76), (19.76)} 0

and

 S ∩ (Γr ∩ I  ) × R)

(19.85)

σ satisfies for all (λ, λ ) ∈ I  the bounds with I  ∈ Λr . By (19.83) and (†), TM,Q 0 c3  (19.76) and (19.76 ) for σ outside a set of measure e−M . The set (19.85) is  3 semialgebraic of degree at most (M A(r ) )C < exp exp(log log r)4 . Take C6 s.t. c3 1+ also c3 C6 > 1. Hence eM > e(log N ) > Ar , and we may carry out the argument (18.29) and (18.35) (with r0 replaced by r ). This provides a collection Λr+1 of C size A−(r+1) intervals I  in R2b , refining {I ∈ Λr }, on which TN−1 , N = Ar+1 , satisfies the required bounds. By (19.81), we have, moreover,    mes b ( (Γr ∩ I)\ (Γr ∩ I  ) < [exp exp(log r)1/3 ]−1 (19.86) I∈Λr

I  ∈Λr+1

Estimate (19.86) is weaker than (18.35) but equally suffices. Proof of Lemma 19.10. Fix B. We need to show that if n1 , . . . , nk ∈ Zd is a sequence of distinct elements s.t. |nj − nj+1 | + |n2j − n2j+1 | < B for 1 ≤ j < k

(19.87)

then k < B , for some C = C(d). Denote C

Δj n = nj+1 − nj Assume I ⊂ [1, . . . , k] an interval s.t. d1 = dim[Δj n|j ∈ I  ] = dim[Δj n|j ∈ I]

(19.88)

APPLICATION TO THE CONSTRUCTION OF QUASI-PERIODIC SOLUTIONS

157

for any subinterval I  ⊂ I with

|I  | ≥ K Our aim is to obtain a lower bound on K in terms of |I| (and B, d). It follows from (19.87) that for |j − j  | ≤ K, |nj − nj  | + |n2j − n2j  | < KB 2|(nj  − nj ).nj | ≤ |n2j − n2j  | + |nj  − nj |2 < 2K 2 B 2 Hence |Δj  n.nj | < 2K 2 B 2 (19.89) Fix j ∈ I, and let j ∈ I ⊂ I, where I  is an interval of length K. Thus (19.88) holds. Let ξ1 , . . . , ξd1 be linearly independent elements in {Δj  n|j  ∈ I  }. Thus [ξ1 , . . . , ξd1 ] = [Δj  n|j  ∈ I  ] = [Δj  n|j  ∈ I] d1 Let ξ = s=1 cs ξs ∈ [ξ1 , . . . , ξd1 ]. Then d1  cs .ξs .ξs = ξ.ξs (1 ≤ s ≤ d1 ) 

s=1

and hence, by Cramer’s rule, | det[(ξs .ξs )1≤s ≤d1 (s =s) , (ξ.ξs )1≤s ≤d1 ]| |cs | = | det[(ξs .ξs )1≤s ≤d1 ,1≤s ≤d1 ]| 1

1

d1

< d12 .|ξ|.B(d12 .B 2 )d1 −1 = d12 B 2d1 −1 |ξ| since |ξs | < B and det[ξs .ξs ] ∈ Z\{0}. By (19.89), |nj .ξs | < 2K 2 B 2 for s = 1, . . . , d1 and (19.90) implies d1  +1 |cs | |ξs .nj | < 2d12 B 2d1 +1 K 2 |ξ| |ξ.nj | ≤

(19.90)

(19.91)



Thus (19.91) holds for all j ∈ I, ξ ∈ [Δj  n|j ∈ I]. Therefore, if j1 , j2 ∈ I, we get d1

+1

(19.92) |nj1 − nj2 | < 4d12 B 2d1 +1 K 2 d Since {nj } consists of distinct elements in Z , we may choose j1 , j2 ∈ I such that 1

|nj1 − nj2 | > |I| d Hence, from (19.92), 1 1 − d+2 −d− 1 2 |I| 2d d 4 B (19.93) 2 This means that there is an interval I  ⊂ I for which 1 |I  | > d−(d+1) B −d−1 |I| 2d and dim[Δj n|j ∈ I  ] < d1 = dim[Δj n|j ∈ I] Starting from I = [1, . . . , k], iteration of the preceding at most d times implies that   −1 −2 1 d 1 d 1 > (dB)−(d+1) 1+(2d) +(2d) +··· k ( 2d ) > (dB)−2(d+1) k ( 2d ) and hence d+1 d+1 k < d(2d+2) B (2d+2) This proves the lemma with Cd = (2d + 2)d+2 .

K>

Chapter Twenty Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations 1. Formulation of the Problem Consider NLW with periodic boundary condition (x ∈ Td ) of the form ytt − Δy + εF  (y) = 0

(20.1)

where F (y) is a polynomial in y. The construction of time-periodic solutions was achieved in arbitrary dimension d (see [B1]). Quasi-periodic solutions were so far only produced for d = 1 (see [B2], [Kuk], and [Wa] with 1D Dirichlet bc). In this chapter we will indicate how the methods described in the preceding chapter may be used to treat this problem in general dimension d. Rather than relying on amplitude-frequency modulation, extracting parameters from the nonlinearity, we discuss again nonlinear perturbations of a linear equation with parameters. Thus replace (20.1) by ytt + B 2 y + εF  (y) = 0

(20.2)

where B is given by a Fourier multiplier defined as follows: Let 0 ∈ {n1 , . . . , nb } ⊂ Zd be a set of distinguished modes. Let B be defined by the multiplier {μn }n∈Zd , where $ (1 ≤ j ≤ b) μnj = λj > 0 (20.3) μn = |n| if n ∈ {n1 , . . . , nb } and where again λ = (λ1 , . . . , λb ) is a b-dim parameter. Introducing as usual the speed v = yt , (20.2) becomes $ yt = v vt = −B 2 y − εF  (y) Denoting z = B −1 v and q = y − iz, we thus obtain $ yt = Bz zt = −By − εB −1 F  (y) or 1 ∂H qt = Bq + εB −1 F  (Re q) = B −1 i ∂ q¯ where    1 H(q) = |Bq|2 + 2εF (Re q) 2

(20.4)

(20.5)

(20.6)

160

CHAPTER 20

Denoting

 H1 (q, q¯) = 2F

q + q¯ 2



the equation 1 ∂H1 qt = Bq + εB −1 i ∂ q¯

(20.7)

will be treated following the method used in the previous chapter for NLS. Write  q(x) = qn e2πin.x n∈Zd

and



qn (t) =



qn (k)eik.λ t

k∈Zb

Let R = {(nj , ej )|j = 1, . . . , b} ⊂ Rd+b be the set of resonant sites and specify again q nj (ej ) = aj ∈ R+

(20.8)

(the case {aj }1≤j≤b ⊂ C may be reduced to (20.8) by time shift). Identifying q and qˆ = qˆ(n, k) = qˆn (k), rewrite (20.7) as (k.λ − μn )ˆ q (n, k) + ε

∂H1 1  (n, k) = 0 μn ∂ q¯

(20.9)

where Bn = μn = 0, and make again a decomposition in P and Q-equations. Thus the Q-equations are λj − λj +

∂H1 ε 1  (nj , ej ) = 0 aj λj ∂ q¯

(1 ≤ j ≤ b)

(20.10)

(which permit us to express λ = λ (λ)), and the P equations are now given by ⎧ ⎨μ (k.λ − μ )u(n, k) + ε ∂H1 n n ∂ q¯ (n, k) = 0 (20.11) ∂H1 ⎩μ (−k.λ − μ )v(n, k) + ε (n, k) = 0 −n

−n

∂q

In the present case, the linearized operator T is T = Tq = D + εSq where D±,n,k = μ±n (±k.λ − μ±n )

(20.12)

We let λ = λ1 , . . . , λb ) vary in an interval Ω ⊂ Rb , and ε is taken small enough. In this setting, we have the analogue of Theorem 19.1 for (20.7).

161

CONSTRUCTION OF QUASI-PERIODIC SOLUTIONS

The proof follows the same scheme explained in Chapters 18 and 19. There are some additional ingredients and issues that we will explain next. 2. Technical Details Compared with the NLS case discussed in the previous chapter, the inversion of σ the TM,Q operators requires certain modification of the arguments. In the present situation, σ D±,n,k = μn (±(k.λ + σ) − μn )

(20.13)

where μn = |n| = 0 except for n ∈ {nj |j = 1, . . . , b}. The most significant issue is to prove Lemma 19.13 in the present situation. This relies on the arithmetical Lemma 19.10, which is explicitly dependent on properties of the {μn }. To obtain the required separation result, an additional nonlinear condition on λ will be imposed. The next fact will substitute Lemma 19.10. Lemma 20.14. Let B be a large number and assume λ ∈ Rb satisfi es the condition |P (λ )| > B −C



for all polynomials P (X) ∈ Z[X1 , . . . , Xb ], P (X) = 0 of degree deg P < 10d and with coeffi cients |aα | < B C . Consider a sequence (ξj )1≤j≤k of distinct elements of Zb+d s.t. for some σ ∈ R, for all j |(λ .kj + σ)2 − |nj |2 | < 1 where ξj = (kj , nj )

(20.15)

|ξj − ξj−1 | < B

(20.16)

max(#{1 ≤ j ≤ k|nj = n}) < B 

(20.17)

and Assume, moreover, that n

Then k < (BB  )C



(20.18)

(We will denote in what follows all constants possibly dependent on d and b by C.) We prove Lemma 20.14 at the end of this chapter. Lemma 20.19. Lemma 19.13 holds provided λ ∈ Rb satisfi es, moreover, the property |p(λ )| > M0−C



(20.20)

for any polynomial p(X) ∈ Z[X1 , . . . , Xb ], p(X) = 0 of degree at most 10d and coeffi cients {aα } bounded by M0C . Sketch of the Proof. We proceed again by multiscale analysis in M 0 . Assume σ that the statement holds at scale M0 for TM , where Q ⊂ Zd is a M0 -interval, 0 ,Q minn∈Q |n| > CM0 . 1c 2 2

Then take M1 = eM0 and let Q ⊂ Zd be an M1 -interval s.t. minn∈Q |n| > σ+k.λ CM1 . Call (n, k) a good site if TM , Q0 = [−M0 , M0 ]d satisfies (19.15) and 0 ,n+Q0 (19.15). Otherwise, (n, k) is called a bad site.

162

CHAPTER 20

Partition [−M1 , M1 ]b × Q in (b + d)-dim intervals P of size M0 . Consider the collection P of intervals P containing some bad site (n, k). Thus [−M1 , M1 ]b × Q ≡ Ω = Ω0 ∪ Ω1 where Ω1 =



P

P ∈P

and (TΩσ0 )−1 is controlled by the resolvent identity. Our next goal is to show that Ω1 has a separated cluster structure. Fix B = M1ρ , ρ small enough Let {Pi |i = 1, . . . , j} be a sequence of distinct elements in P s.t. dist(Pi , Pi+1 ) < B Clearly, if ξ is a bad site, there is some ξ  = (n , k  ) ∈ [−M1 , M1 ]b+d for which < M |ξ − ξ  | ∼ 0

and σ min |D±,n  ,k | < ±

Hence

1 3

  (k .λ + σ)2 − |n |2 < 1

We therefore may find a sequence ξi = (ni , ki ), 1 ≤ i ≤ j, s.t. dist (ξi , Pi ) < CM0 and hence |ξi+1 − ξi | < B + CM0 < 2B and

(ki .λ + σ)2 − |ni |2 < 1

In order to apply Lemma 20.14, we need to check condition (20.17). By construction, it is clear that given n ∈ [−M1 , M1 ]d , #{1 ≤ i ≤ j ni = n}  < (CM0 )d maxn [#{k ∈ [−M1 , M1 ]b T σ+k.λ fails (19.15), (19.15)}]  M0 ,n +Q0

(20.21)

P n n

..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .......• . ... . ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ....... ..... ..... ..... ..... ..... ..... ..... ..... ..... . . ... . ... . ... . ... . ... . ... .

k

............ .........

Since in particular from the assumption, λ satisfies the DC k.λ  > M1−C for k ∈ Zb , 0 < |k| < M1

163

CONSTRUCTION OF QUASI-PERIODIC SOLUTIONS

it follows again from the validity of Lemma 20.19 at scale M 0 that #{. . .} < e(log M0 )

C

Hence (20.21) < (CM0 )d e(log M0 )

C

and we may take B  = e(log M0 )

C

in (20.17). Applying Lemma 20.14, we conclude that 1  C C < M1ρC+ < M1100 j < B.e(log M0 ) for appropriate choice of ρ. This proves that Ω1 =



Ω1,β

β

where 1

diam Ω1,β < M1100 and dist (Ω1,β , Ω1,β  ) > M1ρ for β = β  3

1,β be an M 4 ρ We continue the argument as in Lemma 19.13. Let thus Ω 1 neighborhood of Ω1,β . Assume σ > 0. Since minn∈Q |n| > CM1 , clearly,   σ | = |n| k.λ + σ + |n| > M1 CM1 − 0(M1 ) > M12 |D−,n,k and (R− TΩσ

1,β

R− )−1  < M1−2

The {σi }-functions are introduced as in Lemma 19.13. Thus, if ρ/2

min |σ − σi (λ, λ )| > e−M1 i

we get (TΩσ

ρ/2

1,β

)−1  < eM1

for all β

The resolvent identity then gives the desired bound on (T Ωσ )−1 . The proof of the analogue of Lemma 19.38 is essentially the same. The only difference is that (19.49) gets replaced by the (weaker) restriction |k − ¯k | + |n| − R < M 1+ and (19.56) becomes #Λ < M C .Rd−1

164

CHAPTER 20 1 100d

The argument based on the dichotomy R < M1 The remainder of the argument is the same.

1 100d

, R > M1

remains identical.

3. Proof of Lemma 20.14. Since, by density, we may take σ = k.λ for some k ∈ Zb and the problem is invariant under translation, we may assume σ = 0. Thus the hypothesis (20.15) becomes |(λ .kj )2 − |nj |2 < 1 (1 ≤ j ≤ k) (20.22) Define the operators T± : Rb+d → Rd+1 : (k, n) → (n, ±k.λ ) Thus, for ξ = (k, n), |n|2 − (λ .k)2 = T+ ξ.T− ξ The hypothesis on λ is exploited as follows. Lemma 20.23. Let (ξs )1≤s≤d1 (d1 ≤ d), ξs = (ks , ns ) ∈ Zb+d , |ξs | < B , s.t. dim[ξs 1 ≤ s ≤ d1 ] = d1

and

dim[ns 1 ≤ s ≤ d1 ] ≥ d1 − 1

Then, under the assumption on λ , we have

| det(T+ ξs .T+ ξs )1≤s,s ≤d1 > B −C

(20.24)

| det(T+ ξs .T− ξs )1≤s,s ≤d1 | > B −C

(20.24 )

and

Proof of Lemma 20.22. We prove (20.24). Define   P (X) = det ns .ns − (ks .X)(ks .X) ∈ Z[X1 , . . . , Xb ] which is a polynomial of degree ≤ 2d1 and coefficients  d1 < CB 2d1 |aα | ≤ d1 !(1 + b2 )d1 |ξs |2

(20.25)

We claim that P (X) = 0. If P (X) = 0, then also   0 = P (iX) = det ns .ns + (ks .X)(ks .X) implying that the vectors (ns , ks .X) ∈ Rd+1 (1 ≤ s ≤ d1 ) are linearly dependent for all X ∈ Rb . Since from assumption dim[ns 1 ≤ s ≤ d1 ] = d1 − 1 there is a unique (up to multiples) vector (cs )1≤s≤d1 ∈ Rd1 \{0} s.t.  cs ns = 0 in Rd

CONSTRUCTION OF QUASI-PERIODIC SOLUTIONS

165

and therefore also satisfying    cs ks = 0 in Rb cs ks .X = 0 hence This contradicts the linear independence of (ξs )1≤s≤d1 . Thus P (X) = 0 and (20.24) follow from (20.25) and the assumption on λ . We now return to the proof of Lemma 20.14. Denote Δj ξ = ξj − ξj−1 

Fix K ∈ Z+ and let |j − j | ≤ K. From (20.16) and (20.22) 2|T+ ξj .T− (ξj  − ξj )|

≤ |T+ ξj .T− ξj | + |T+ ξj  .T− ξj  | +|T+ (ξj  − ξj ).T− (ξj  − ξj )| ≤ 2 + Cξj  − ξj 2 < CK 2 B 2

Hence, by subtraction, also |T+ ξj .T− Δj  ξ| < CK 2 B 2

(20.26)

Let I ⊂ [1, k] be an interval and K ∈ Z+ s.t. d1

≡ dim[T+ Δj ξ j ∈ I] = dim[T+ Δj ξ j ∈ I  ]

for all I  ⊂ I, |I  | ≥ K. Then take j ∈ I, and let I  , |I  | = K be an interval s.t. j ∈ I  ⊂ I. Denote E± = [T± Δj  ξ j  ∈ I  ] = [T± Δj  ξ|j  ∈ I] and {e1 , . . . , ed1 } ⊂ {T+ Δj  ξ|j  ∈ I  } a basis for E+ . By (20.26) and the choice of I  , |T− ξj .es | < CK 2 B 2

(1 ≤ s ≤ d1 )

(20.27)



d+1 Since the vectors {es = (Δ js n, Δjs k.λ )}1≤s≤d1 are linearly independent in R , it follows that {Δjs ξ 1 ≤ s ≤ d1 } are linearly independent and dim[Δjs n 1 ≤ s ≤ d1 ] ≥ d1 − 1. Since Δjs ξ < B, (20.24) implies that

| det(es .es )|

= | det(T+ Δjs ξ.T+ Δjs ξ)| > B −C

(20.28)

From (20.28) and Cramer’s rule, BC 

d1 

cs es  ≥ max |cs | for all {cs }

s=1

and hence, by (20.27), |T− ξj .v| < B C K 2

(20.29)

for any v ∈ E+ , v ≤ 1. Equivalently, |T+ ξj .w| < B C K 2

(20.30)

166

CHAPTER 20

for any w ∈ E− , w ≤ 1. Recall that j ∈ I was an arbitrarily chosen element. Let ζ = ξj − ξj  for j, j  ∈ I. It follows from (20.30) that |T+ ζ.w| < B C K 2

(20.31)

for any w ∈ E− , w ≤ 1. Since T+ ζ ∈ E+ , we may write T+ ζ =

d1 

cs e s

s=1

 cs es .¯ es = |T+ ζ.¯ es | < B C K 2

(20.32)

where e¯s = T− Δjs ξ. By (20.24’), also | det(es .¯ es )| = | det(T+ Δjs ξ.T− Δjs ξ)| > B −C

(20.33)

From (20.32) and (20.33) and Cramer, |cs | < B C K 2 Hence nj − nj   ≤ T+ (ξj − ξj  ) = T+ ζ < B C K 2 It follows that Hence

(20.34)

diam {nj j ∈ I} < B C K 2 #{nj j ∈ I} < B C K 2d

(20.35)

Since we also assume (20.17), (20.35) implies that |I| < B  B C K 2d

(20.36)



This means that there is an interval I ⊂ I satisfying 1

1

|I  | > (B  )− 2d B −C |I| 2d for which

(20.37)

dim[T+ Δj ξ j ∈ I  ] < dim[T+ Δj ξ j ∈ I]

Starting from I = [1, k], at most d + 1 iterations of the preceding clearly lead to the following conclusion: 1

1

1 > (B  )− d B −C k ( 2d ) This proves (20.18) and Lemma 20.14.

d

(20.38)

CONSTRUCTION OF QUASI-PERIODIC SOLUTIONS

167

References [1] J. Bourgain. Construction of periodic solutions of nonlinear wave equations in higher dimension, GAFA 5 (1995), 629–639. [2] J. Bourgain. Nonlinear Schrödinger equations. In Hyperbolic equations and frequency interactions, IAS/Park City Math. Ser. 5, Amer. Math. Soc., Providence, RI, 1999, pp. 3–157. [3] S. Kuksin. Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Math., Springer-Verlag, Berlin, 1993, p. 1556. [4] C. Wayne. Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, CMP 127 (1990), 479–528.

Appendix Strongly Mixing Potentials We first recall the Figotin-Pastur [F-P] formalism, based on the representation of Schrödinger matrices in polar coordinates. Consider the discrete Schrödinger operator (1)

(Hλ ψ)n = ψn+1 + ψn−1 + λvn ψn

where vn is a sequence of real numbers and λ > 0. We consider (1) both on the integer lattice Z or on the half-line Z+ ∪ {0}. At this stage, we do not specify {vn }. Fix some δ > 0, and restrict the energy to δ < |E| < 2 − δ

(2)

Then define κ ∈ (0, π) and Vn by E = 2 cos κ vn Vn = − sin κ Next, let ψ be a solution of Hψ = Eψ on the half-line Z+ ∪ {0}. Thus      ψn E − λvn −1 ψn+1 = for any n = 1, 2, . . . . 1 0 ψn ψn−1

(3) (4)

(5)

The coordinate change Yn = (ψn − cos κψn−1 , sin κψn−1 ) transforms (5) into  cos κ Yn+1 = sin κ

− sin κ cos κ



 Yn + λVn

sin κ cos κ 0 0

(6)  Yn

(7)

Introducing polar coordinates, Yn = ρn (cos ϕn , sin ϕn ) Equation (7) implies     cos ϕn+1 cos(ϕn + κ) + λVn sin(ϕn + κ) = ρn ρn+1 sin(ϕn + κ) sin ϕn+1 cotgϕn+1 = cotg (ϕn + κ) + λVn  2

ρ2n+1 = ρn



  1 + λVn sin 2(ϕn + κ) + λ2 Vn2 sin2 (ϕn + κ)

(8)

(9) (10) (11)

170

APPENDIX

From (11) with ρ1 = 1, we get N  1  1 log(1 + λVn sin 2(ϕn + κ) + λ2 Vn2 sin2 (ϕn +κ) (12) log ρN (θ) = N 2N 1

=

N λ2  2 V 8N 1 n

(13)

+

N λ  Vn sin 2(ϕn + κ) 2N 1

(14)

N λ2  2 − V cos 2(ϕn + κ) 4N 1 n

+

N λ2  2 V cos 4(ϕn + κ) + O(λ3 ) 8N 1 n

(15) (16)

Letting dζn = e2iϕn μ = e2iκ

(17)

one verifies that (10) is equivalent to ζn+1 = μζn +

(μζn − 1)2 iλ Vn 2 1 − iλ 2 Vn (μζn − 1)

(18)

The idea is that (13) produces the main term and (14) to (16) appear as error terms for N → ∞ (assuming the potential {vn } sufficiently mixing). Notice that (18) permits us to recover the phases {ϕn } recursively, but tracking their distribution from this formula is not obvious. Figotin and Pastur [F-P], Theorem 14.6, used this formalism to show that for small λ the Lyapounov exponent L(λ, E) obeys the expansion λ2 E(v02 ) + O(λ3 ) (19) 2(4 − E 2 ) provided the potentials are identically distributed independent random variables with zero mean (notice that a change of variables, as the one given by (6), does not change the limit limN →∞ N1 log(|ψn−1 | + |ψn |) = L(E).) The constant in O(λ3 ) depends on E but remains bounded on intervals of the form [−2 2 − δ]. Observe  + δ, −δ]∪[δ,  that ζn+1 , hence ϕn+1 , only depends on V1 , . . . , Vn by (18) . In particular, if {vn } are i.i.d., Vn and ϕn are independent random variables. Taking expectations in (13) to (16), one thus obtains (19). Suppose next that vn = vn (x) = F (T n x), where T : X → X is an ergodic transformation on some probability space (X, μ). If T is assumed strongly mixing, Chulaevsky and Spencer showed in [C-S] that (19) remains valid–with E(v 02 ) replaced by the expression ∞  σ(E) = e2iκ F, F (T  .) (20) L(λ, E) =

=−∞

APPENDIX

171

This is accomplished by iteration of (18) and exploiting the decay of the correlations. Typical examples are (i) The Period Doubling Map. Thus v = λF (2n x) (x ∈ T) (21)

n where F is 2π-periodic and F = 0. (ii) Hyperbolic Toral Automorphisms A : T2 → T2 . In this case, vn = λF (An x), x ∈ T2 (22)

and again F is 2π-periodic, and F = 0. In both examples (i) and (ii), we assume F a C 1 -function (this suffices for our purpose). In these cases it was shown in [C-S] that L(λ, E) admits an asymptotic expansion L(λ, E) = λ2 c0 (E) + o(λ3 ) (23) for small λ and 0 < |E| < 2. Here c0 (E) is some function of E that depends on F . In [B-G] we basically develop further the [C-S] analysis. Returning to (14) to (16), it turns out that (14) is the most difficult term to control (because it is linear in λ). Now in examples (21) and (22), it is possible with some work to control this sum by basically martingale difference sequences. Using the standard deviation estimates in martingale theory, this allows us to obtain an LDT 1 mes [x| log MN (x, E) − LN (E) > λ5/2 ] < exp(−Cλ n) (24) N Combining (24) with some of the methods developed in the quasi-periodic case, in particular in Chapters 6, 7, and 10, the following further results are obtained in [B-S]. Theorem. Let Hλ (x) = λF (T n x)δnn + Δ with T and F as in examples (21) and (22). Fix the energy range δ < |E| < 2 − δ , and let λ > 0 be small enough. Then the IDS of Hλ is Hölder continuous. Moreover, in example (21), Hλ (θ) satisfi es Anderson localization (i.e., pure point spectrum with exponentially localized states) for almost all θ. In example (22), AL holds on any interval I0 ⊂ [−2 + δ, −δ] ∪ [δ, 2 − δ] on which σ(E) > 0, where σ(E) is given by (20). See [B-S] for further details. Remarks. 1. The Figotin-Pastur method uses the fact that λ is small. In analogy with the random models, one should expect the theorem to hold for arbitrary λ = 0 (considering, for instance, the model (21) . It does not seem obvious, however, to adjust the Furstenberg-Lepage technique based on Perron-Frobenius to a quasi-random setting. 2. In the same spirit, one may ask whether the IDS for the operator H λ considered above is smooth. 3. One could hope that the Figotin-Pastur method also may be applicable in the context of weakly mixing transformations, for instance, given by a skew shift. Thus take n(n − 1) Hλ (x) = λ cos(x1 + nx2 + ω)δnn + Δ (25) 2

172

APPENDIX

Can one use the Figotin-Pastur approach to prove positivity of the Lyapounov exponent of (25) for λ > 0? Some results on Lyapounov exponent and localization of (25) for small λ may be found in [B1] and [B2].

APPENDIX

173

References [B1] J. Bourgain. Positive Lyapounov exponents for most energies, GAFA 1745 (2000), 37–66. [B2] J. Bourgain. On the spectrum of lattice Schrödinger operators with deterministic potential, J. Analyse 87 (2002), 37–75 and 88 (2002), 221–254. [B-S] J. Bourgain, W. Schlag. Anderson localization for Schrödinger operators on Z with strongly mixing potentials, CMP 215 (2000), 143–175. [C-S] V. Chulaevsky, T. Spencer. Positive Lyapounov exponents for a class of deterministic potentials, CMP 168 (1995), 455–466. [F-P] A. Figotin, L. Pastur. Spectra of random and almost periodic operators, Grundlehren der mathematischen Wissenshaften, Springer, Berlin 1992, p. 297.