Products of Random Matrices with Applications to Schrödinger Operators

1. Front Matter

   Pages i-x

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2. Limit Theorems for Products of Random Matrices

   1. Front Matter

      Pages xi-4

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   2. The Upper Lyapunov Exponent

      • Philippe Bougerol, Jean Lacroix

      Pages 5-15

   3. Matrices of Order Two

      • Philippe Bougerol, Jean Lacroix

      Pages 17-42

   4. Contraction Properties

      • Philippe Bougerol, Jean Lacroix

      Pages 43-76

   5. Comparison of Lyapunov Exponents and Boundaries

      • Philippe Bougerol, Jean Lacroix

      Pages 77-99

   6. Central Limit Theorem and Related Results

      • Philippe Bougerol, Jean Lacroix

      Pages 101-144

   7. Properties of the Invariant Measure and Applications

      • Philippe Bougerol, Jean Lacroix

      Pages 145-171

3. Back Matter

   Pages 173-180

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4. Random Schrödinger Operators

   1. Front Matter

      Pages 181-186

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   2. The Deterministic Schrödinger Operator

      • Philippe Bougerol, Jean Lacroix

      Pages 187-203

   3. Ergodic Schrödinger Operators

      • Philippe Bougerol, Jean Lacroix

      Pages 205-236

   4. The Pure Point Spectrum

      • Philippe Bougerol, Jean Lacroix

      Pages 237-251

   5. Schrödinger Operators in a Strip

      • Philippe Bougerol, Jean Lacroix

      Pages 253-274

5. Back Matter

   Pages 275-284

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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.

• ACE
• THE
• approximation
• difference equation
• differential equation
• distribution
• equation
• form
• function
• matrices
• measure
• operator
• proof
• schrödinger operator
• stochastic differential equation
• matrix theory
• partial differential equations

• UER de Mathématiques, Université Paris 7, Paris, France

  Philippe Bougerol

• Département de Mathématiques, Université de Paris XIII, Villetaneuse, France

  Jean Lacroix

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