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

by Jean Bourgain

Overview

This book presents an overview of recent developments in the area of localization for quasi-periodic lattice Schrödinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. The physical motivation of these models extends back to the works of Rudolph Peierls and Douglas R. Hofstadter, and the models themselves have been a focus of mathematical research for two decades. Jean Bourgain here sets forth the results and techniques that have been discovered in the last few years. He puts special emphasis on so-called "non-perturbative" methods and the important role of subharmonic function theory and semi-algebraic set methods. He describes various applications to the theory of differential equations and dynamical systems, in particular to the quantum kicked rotor and KAM theory for nonlinear Hamiltonian evolution equations.


Intended primarily for graduate students and researchers in the general area of dynamical systems and mathematical physics, the book provides a coherent account of a large body of work that is presently scattered in the literature. It does so in a refreshingly contained manner that seeks to convey the present technological "state of the art."

Product Details

ISBN-13:	9780691120980
Publisher:	Princeton University Press
Publication date:	11/21/2004
Series:	Annals of Mathematics Studies , #158
Pages:	200
Product dimensions:	6.00(w) x 9.25(h) x (d)

About the Author

Jean Bourgain is Professor of Mathematics at the Institute for Advanced Study and J. Doob Professor of Mathematics at the University of Illinois, Urbana-Champaign. He is the author of Global Solutions of Nonlinear Schrödinger Equations.

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Green's Function Estimates for Lattice Schrödinger Operators and Applications

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By Jean Bourgain

Princeton University Press

Copyright © 2004 Princeton University Press
All right reserved.
ISBN: 978-0-691-12098-0

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Chapter One

Introduction

We will consider infinite matrices indexed by Z (or [Z.sup.b]) associated to a dynamical system in the sense that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where x [member of] [OMEGA], and T is an ergodic measure-preserving transformation of [OMEGA]. Typical settings considered here are

[OMEGA] = T Tx = x + w (1 - frequency shift) [OMEGA] = [T.sup.d] Tx = x + w (d - frequency shift)

[OMEGA] = [T.sup.2] Tx = ([x.sub.1] + [x.sub.2], [x.sub.2] + w)(skewshift)

[OMEGA] = [T.sup.2] Tx = Ax, where A [member of] S[L.sub.2] (Z); hyperbolic

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0)

where the [[empty set].sub.k] are functions on [OMEGA].

We will usually assume that H(x) is self-adjoint, although many parts of our analysis are independent of this fact. Define

[H.sub.N] =[R.sub.[1,N]] [H R.sub.[1,N]]

where [R.sub.[1,N]] = coordinate restriction to [1, N][subset] Z; and the associated Green's functions are

[G.sub.N](E) = [([H.sub.N] - E).sup.-1]

(if [H.sub.N] - E is an invertible N × N matrix).

One of our concerns will be to obtain a 'good bound' on [G.sub.N](E, x), except for x in a "small" exceptional set. A typical statement would be the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

for all x outside a set of measure < [e.sup.-N]sigma]] . Here [delta], [sigma] > 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.

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

The spatial variable x [member of] [T.sup.d] (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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.4)

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 [member of] [T.sup.d] for all time. By the reality of V, there is conservation of the [L.sup.2]-norm.

If [u.sub.0] = u(0) [member of] [H.sup.s]([T.sup.d]), then

u(t) [member of] [H.sup.s] for all time

Problem. Possible growth of [[parallel]u(t)]parallel].sub.[H.sup.s]].

Remark 1. It turns out that in (1.4) with typical values of a, b there is almost-periodicity in the following sense: Assume [u.sub.0] sufficiently smooth (depending on s). Then u(t) is almost as periodic as an [H.sup.s]-valued function and, in particular, [sup.sub.t] [parallel]u(t)]parallel][H.sup.s] < [infinity].

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 [L.sup.2]-valued function. But there are examples where V is smooth in x and t and such that for some smooth initial data [u.sub.0]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all s > 0

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 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

i[u.sub.t] + [DELTA]u + [epsilon]F (u, [bar.u]) = 0 (1.5)

The linearized operator expressed in Fourier modes then becomes

T = D + [epsilon]S

where D is diagonal with diagonal elements of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

and S is a Toeplitz-type matrix with (very) smooth symbol, i.e.,

S((k, n), (k', n'))= [??](k - k', n - n')

where [??]([xi]) decays rapidly for |[xi]| [right arrow] [infinity].

In (1.6), b = dimension of invariant tori, and w [member of] [R.sup.b] is the frequency vector. The matrix T is finite (depending on the iteration step), and we seek appropriate bounds on [T.sup.-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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

where [DELTA] is the usual lattice Laplacian

[DELTA] (n, n') = 1 if |n - n'| = 1

= 0 otherwise

Letting v(x) = cos x on T, Tx = x + w = shift, we obtain the Almost Mathieu operator

[H.sub.[lambda]](x) = [lambda] cos(x + nw) + [DELTA] (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 w satisfies a diophantine condition

dist (kw, 2[pi]Z) = [parallel]k, w]parallel] > c [|k|.sup.-c] for k [member of] Z\{0}

Then, for a.e. x,

(i) [lambda] > 2: H(x) has p.p. spectrum

(ii) [lambda] = 2: H(x) has purely s.c. spectrum

(iii) [lambda] < 2: H(x) has purely a.c. spectrum

Thus there is a phase transition at [lambda] = 2.

This model has a special and remarkable self-duality property (wrt Fourier transform) cos [right arrow] 1/2 [DELTA]

[DELTA] [right arrow] 2 cos

observed and exploited first by Aubry. One of its implications is that

Spec [H.sub.[lambda]] = Spec [H.sub.4]/[lambda]

(referring to the "topological spectrum" that is independent of x).

In more general situations involving shifts,

[lambda]v(x + nw)[[delta].sub.nn'] + [DELTA] (1.9)

with v real analytic on [T.sup.d], a rough picture is the following:

[lambda] large: p.p. spectrum with Anderson localization

[lambda] small: purely a.c. spectrum

[lambda] intermediate: possible coexistence of different spectral types

Recall that Anderson localization means the following:

Assume [psi] an extended state, i.e.,

H [psi] = E [psi] and |[[psi].sub.n]| [??] [|n|.sup.c]

Then [psi] [member of] [l.sup.2] and

|[[psi].sub.n]| < [e.sup.-c|n|] for |n| [right arrow] [infinity]

(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 = ([lambda] cos n[w.sub.1] + [tau] cos n[w.sub.2])[[delta].sub.nn'] + [DELTA] (1.10)

where [lambda] < 2, and [tau] is small. Then, for w = ([w.sub.1], [w.sub.1]) 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 [lambda] > 0 (as is the case of a random potential).

This problem is open at this time. It is known that for all [lambda] > 0 and w in a set of positive measure

H = [lambda](cos n(n - 1) / 2 w)[[delta].sub.nn'] + [DELTA]

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 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 nonlinearwave equations (NLW). Earlier research, mainly due to C. Wayne, S. Kuksin, W. Craig, and myself (see 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 general 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

[H.sub.x] = [lambda]v(x + nw) + [DELTA] (1.11)

where v is a real analytic potential on [T.sup.d] (d = 1 or d > 1), and [DELTA] 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 [lambda]. The results are nonperturbative in the sense that the condition [lambda] > [[lambda].sub.0](v) depends on v only and not on the arithmetical properties of the rotation vector w (provided we assume w 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 [lambda]. 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 [Z.sup.2]-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 matrix-valued functions.

(Continues...)

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Table of 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 Function 31
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 Z ^d -lattice ( d > 1) 123
CHAPTER 18: An Approach to Melnikov's Theorem on Persistency of Non-resonant Lower Dimension Tori 133
CHAPTER 19: Application to the Construction of Quasi-Periodic Solutions of Nonlinear Schr dinger Equations 143
CHAPTER 20: Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations 159
Appendix 169

Editorial Reviews

"This text is an up to date introduction to localization problems for lattice Schrödinger operations with deterministic ergodic potentials by one of the leading experts. . . . I can recommend it to any graduate student or researcher in the field."—-G. Teschl, Monatschefte für Mathematik

From the Publisher

This text is an up to date introduction to localization problems for lattice Schrödinger operations with deterministic ergodic potentials by one of the leading experts. . . . I can recommend it to any graduate student or researcher in the field.

Monatschefte für Mathematik - G. Teschl

This text is an up to date introduction to localization problems for lattice Schrödinger operations with deterministic ergodic potentials by one of the leading experts. . . . I can recommend it to any graduate student or researcher in the field.
— G. Teschl

Monatschefte fur Mathematik

This text is an up to date introduction to localization problems for lattice Schrödinger operations with deterministic ergodic potentials by one of the leading experts. . . . I can recommend it to any graduate student or researcher in the field.
— G. Teschl

Monatshefte fur Mathematik fur Mathematik

This text is an up to date introduction to localization problems for lattice Schrödinger operations with deterministic ergodic potentials by one of the leading experts. . . . I can recommend it to any graduate student or researcher in the field.
— G. Teschl

Monatschefte für Mathematik