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Products of Random Matrices with Applications to Schrödinger Operators

  • Philippe Bougerol
  • Jean Lacroix

Part of the Progress in Probability and Statistics book series (PRPR, volume 8)

Table of contents

  1. Front Matter
    Pages i-x
  2. Limit Theorems for Products of Random Matrices

    1. Front Matter
      Pages xi-4
    2. Philippe Bougerol, Jean Lacroix
      Pages 5-15
    3. Philippe Bougerol, Jean Lacroix
      Pages 17-42
    4. Philippe Bougerol, Jean Lacroix
      Pages 43-76
    5. Philippe Bougerol, Jean Lacroix
      Pages 77-99
    6. Philippe Bougerol, Jean Lacroix
      Pages 101-144
    7. Philippe Bougerol, Jean Lacroix
      Pages 145-171
    8. Back Matter
      Pages 173-180
  3. Random Schrödinger Operators

    1. Front Matter
      Pages 181-186
    2. Philippe Bougerol, Jean Lacroix
      Pages 187-203
    3. Philippe Bougerol, Jean Lacroix
      Pages 205-236
    4. Philippe Bougerol, Jean Lacroix
      Pages 237-251
    5. Philippe Bougerol, Jean Lacroix
      Pages 253-274
  4. Back Matter
    Pages 275-284

About this book

Introduction

CHAPTER I THE DETERMINISTIC SCHRODINGER OPERATOR 187 1. The difference equation. Hyperbolic structures 187 2. Self adjointness of H. Spectral properties . 190 3. Slowly increasing generalized eigenfunctions 195 4. Approximations of the spectral measure 196 200 5. The pure point spectrum. A criterion 6. Singularity of the spectrum 202 CHAPTER II ERGODIC SCHRÖDINGER OPERATORS 205 1. Definition and examples 205 2. General spectral properties 206 3. The Lyapunov exponent in the general ergodie case 209 4. The Lyapunov exponent in the independent eas e 211 5. Absence of absolutely continuous spectrum 221 224 6. Distribution of states. Thouless formula 232 7. The pure point spectrum. Kotani's criterion 8. Asymptotic properties of the conductance in 234 the disordered wire CHAPTER III THE PURE POINT SPECTRUM 237 238 1. The pure point spectrum. First proof 240 2. The Laplace transform on SI(2,JR) 247 3. The pure point spectrum. Second proof 250 4. The density of states CHAPTER IV SCHRÖDINGER OPERATORS IN A STRIP 2';3 1. The deterministic Schrödinger operator in 253 a strip 259 2. Ergodie Schrödinger operators in a strip 3. Lyapunov exponents in the independent case. 262 The pure point spectrum (first proof) 267 4. The Laplace transform on Sp(~,JR) 272 5. The pure point spectrum, second proof vii APPENDIX 275 BIBLIOGRAPHY 277 viii PREFACE This book presents two elosely related series of leetures. Part A, due to P.

Keywords

ACE THE approximation difference equation differential equation distribution equation form function matrices measure operator proof schrödinger operator stochastic differential equation

Editors and affiliations

  • Philippe Bougerol
    • 1
  • Jean Lacroix
    • 2
  1. 1.UER de MathématiquesUniversité Paris 7ParisFrance
  2. 2.Département de MathématiquesUniversité de Paris XIIIVilletaneuseFrance

Bibliographic information