Products of Random Matrices with Applications to Schrödinger Operators

Products of Random Matrices with Applications to Schrödinger Operators

by P. Bougerol, Lacroix



Product Details

ISBN-13: 9780817633240
Publisher: Birkhauser Verlag
Publication date: 10/01/1985
Series: Progress in Probability Series , #8
Pages: 284

Table of Contents

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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