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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.

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A: "Limit Theorems for Products of Random Matrices".- I - The Upper Lyapunov Exponent.- 1. Notation.- 2. The upper Lyapunov exponent.- 3. Cocycles.- 4. The theorem of Furstenberg and Kesten.- 5. Exercises.- II - Matrices of Order Two.- 1. The set-up.- 2. Two basic lemmas.- 3. Contraction properties.- 4. Furstenberg´s theorem.- 5. Some simple examples.- 6. Exercises.- 7. Complements.- III - Contraction Properties.- 1. Contracting sets.- 2. Strong irreducibility.- 3. A key property.- 4. Contracting action on P(?d) and convergence in direction.- 5. Lyapunov exponents.- 6. Comparison of the top Lyapunov exponents and Furstenberg´s theorem.- 7. Complements. The irreducible case.- IV - Comparison of Lyapunov Exponents and Boundaries.- 1. A criterion ensuring that Lyapunov exponents are distinct.- 2. Some examples.- 3. The case of symplectic matrices.- 4. ?-boundaries.- V - Central Limit Theorems and Related Results.- 1. Introduction.- 2. Exponential convergence to the invariant measure.- 3. A lemma of perturbation theory.- 4. The Fourier-Laplace transform near o.- 5. Central limit theorem.- 6. Large deviations.- 7. Convergence to y.- 8. Convergence in distribution without normalization.- 9. Complements: linear stochastic differential equations.- VI - Properties of the Invariant Measure and Applications.- 1. Convergence in the Iwasawa decomposition.- 2. Limit theorems for the coefficients.- 3. Behaviour of the rows.- 4. Regularity of the invariant measure.- 5. An example: random continued fractions.- Suggestions for Further Readings.- B: "Random Schrödinger Operators".- I - The Deterministic Schrodinger Operator.- 1. The difference equation. Hyperbolic structures.- 2. Self adjointness of H. Spectral properties.- 3. Slowly increasing generalized eigenfunctions.- 4. Approximations of the spectral measure.- 5. The pure point spectrum. A criterion.- 6. Singularity of the spectrum.- II - Ergodic Schrödinger Operators.- 1. Definition and examples.- 2. General spectral properties.- 3. The Lyapunov exponent in the general ergodic case.- 4. The Lyapunov exponent in the independent case.- 5. Absence of absolutely continuous spectrum.- 6. Distribution of states. Thouless formula.- 7. The pure point spectrum. Kotani´s criterion.- 8. Asymptotic properties of the conductance in the disordered wire.- III - The Pure Point Spectrum.- 1. The pure point spectrum. First proof.- 2. The Laplace transform on SI(2,?).- 3. The pure point spectrum. Second proof.- 4. The density of states.- IV - Schrödinger Operators in a Strip.- 1. The deterministic Schrödinger operator in a strip.- 2. Ergodic Schrödinger operators in a strip.- 3. Lyapunov exponents in the independent case. The pure point spectrum (first proof).- 4. The Laplace transform on Sp(1,?).- 5. The pure point spectrum, second proof.

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