Products of random matrices with applications to Schrodinger operators

Title Page......Page 3
Copyright......Page 4
CONTENTS......Page 5
Preface......Page 9
PART A LIMIT THEOREMS FOR PRODUCTS OF RANDOM MATRICES......Page 11
INTRODUCTION......Page 13
1. Notation......Page 17
2. The upper Lyapunov exponent......Page 18
3. Cocycles......Page 20
4. The theorem of Furstenberg and Kesten......Page 23
5. Exercises......Page 25
1. The set-up......Page 29
2. Two basic lemmas......Page 31
3. Contraction properties......Page 36
4. Furstenberg's theorem......Page 42
5. Some simple examples......Page 45
6. Exercises......Page 48
7. Complements......Page 50
CHAPTER III - CONTRACTION PROPERTIES......Page 55
1. Contracting sets......Page 56
2. Strong irreducibility......Page 60
3. A key property......Page 62
4. Contracting action on P(]Rd) and convergence in direction......Page 67
5. Lyapunov exponents......Page 72
6. Comparison of the top Lyapunov exponents and Furstenberg's theorem......Page 76
7. Complements. The irreducible case......Page 80
1. A criterion ensuring that Lyapunov exponents are distinct......Page 89
2. Some examples......Page 93
3. The case of symplectic matrices......Page 99
4. p-boundaries......Page 105
1. Introduction......Page 113
2. Exponential convergence to the invariant measure......Page 115
3. A lemma of perturbation theory......Page 123
4. The Fourier-Laplace transform near 0......Page 128
5. Central limit theorem......Page 133
6. Large deviations......Page 141
7. Convergence to gamma_p......Page 146
8. Convergence in distribution without normalization......Page 147
9. Complements : linear stochastic differential equations......Page 152
CHAPTER VI - PROPERTIES OF THE INVARIANT MEASURE AND APPLICATIONS......Page 157
1. Convergence in the Iwasawa decomposition......Page 159
2. Limit theorems for the coefficients......Page 167
3. Behaviour of the rows......Page 171
4. Regularity of the invariant measure......Page 173
5. An example : random continued fractions......Page 178
SUGGESTIONS FOR FURTHER READINGS......Page 185
BIBLIOGRAPHY......Page 187
PART B RANDOM SCHRODINGER OPERATORS......Page 193
INTRODUCTION......Page 195
1. The difference equation. Hyperbolic structures......Page 199
2. Self adjointness of H. Spectral properties......Page 202
3. Slowly increasing generalized eigenfunctions......Page 207
4. Approximations of the spectral measure......Page 208
5. The pure point spectrum. A criterion......Page 212
6. Singularity of the spectrum......Page 214
1. Definition and examples......Page 217
2. General spectral properties......Page 218
3. The Lyapunov exponent in the general ergodic case......Page 221
4. The Lyapunov exponent in the independent case......Page 223
5. Absence of absolutely continuous spectrum......Page 233
6. Distribution of states. Thouless formula......Page 236
7. The pure point spectrum. Kotani's criterion......Page 244
8. Asymptotic properties of the conductance in the disordered wire......Page 246
CHAPTER III - THE PURE POINT SPECTRUM......Page 249
1. The pure point spectrum. First proof......Page 250
2. The Laplace transform on Sl(2,]R)......Page 252
3. The pure point spectrum. Second proof......Page 259
4. The density of states......Page 262
1. The deterministic Schrodinger operator in a strip......Page 265
2. Ergodic Schrodinger operators in a strip......Page 271
3. Lyapunov exponents in the independent case. The pure point spectrum (first proof)......Page 274
4. The Laplace transform on Sp(i.,IR)......Page 279
5. The pure point spectrum, second proof......Page 284
APPENDIX......Page 287
BIBLIOGRAPHY......Page 289