Showing papers by "René Carmona published in 1990"

Spectral Theory of Random Schrödinger Operators

Abstract: I Spectral Theory of Self-Adjoint Operators.- 1 Domains, Adjoints, Resolvents and Spectra.- 2 Resolutions of the Identity.- 3 Representation Theorems.- 4 The Spectral Theorem.- 5 Quadratic Forms and Self-adjoint Operators.- 6 Self-adjoint Extensions of Symmetric Operators.- 7 Problems.- 8 Notes and Complements.- II Schrodinger Operators.- 1 The Free Hamiltonians.- 2 Schrodinger Operators as Perturbations.- 2.1 Self-adjointness.- 2.2 Perturbation of the Absolutely Continuous Spectrum.- 2.3 An Approximation Argument.- 3 Path Integral Formulas.- 3.1 Brownian Motions and the Free Hamiltonians.- 3.2 The Feynman-Kac Formula.- 4 Eigenfunctions.- 4.1 L2-Eigenfunctions.- 4.2 The Periodic Case.- 4.3 Generalized Eigenfunction Expansions.- 5 Problems.- 6 Notes and Complements.- III One-Dimensional Schrodinger Operators.- 1 The Continuous Case.- 1.1 Essential Self-adjointness.- 1.2 The Operator in an Interval.- 1.3 Green's and Weyl-Titchmarsh's Functions.- 1.4 The Propagator.- 1.5 Examples.- 2 The Lattice Case.- 3 Approximations of the Spectral Measures.- 4 Spectral Types.- 4.1 Absolutely Continuous Spectrum.- 4.2 Singular Spectrum.- 4.3 Pure Point Spectrum.- 5 Quasi-one Dimensional Schrodinger Operators.- 5.1 The Schrodinger Operator in a Strip.- 5.2 Approximation of the Spectral Measures.- 5.3 Nature of the Spectrum.- 6 Problems.- 7 Notes and Complements.- IV Products of Random Matrices.- 1 General Ergodic Theorems.- 2 Matrix Valued Systems.- 3 Group Action on Compact Spaces.- 3.1 Definitions and Notations.- 3.2 Laplace Operators on the Space of Continuous Functions.- 3.3 The Laplace Operators on the Space of Holder Continuous Functions.- 4 Products of Independent Random Matrices.- 4.1 The Upper Lyapunov Exponent.- 4.2 The Lyapunov Spectrum.- 4.3 Schrodinger Matrices.- 5 Markovian Multiplicative Systems.- 5.1 The Upper Lyapunov Exponent.- 5.2 The Lyapunov Spectrum.- 5.3 Laplace Transform.- 6 Boundaries of the Symplectic Group.- 7 Problems.- 8 Notes and Comments.- V Ergodic Families of Self-Adjoint Operators.- 1 Measurability Concepts.- 2 Spectra of Ergodic Families.- 3 The Case of Random Schrodinger Operators.- 3.1 Examples.- 4 Regularity Properties of the Lyapunov Exponents.- 4.1 Subharmonicity.- 4.2 Continuity.- 4.3 Local Holder Continuity.- 4.4 Smoothness.- 5 Problems.- 6 Notes and Complements.- VI The Integrated Density of States.- 1 Existence Problems.- 1.1 Setting of the Problem.- 1.2 Path Integral Approach.- 1.3 Functional Analytic Approach.- 2 Asymptotic Behavior and Lifschitz Tails.- 2.1 Tauberian Arguments.- 2.2 The Anderson Model.- 3 More on the Lattice Case.- 4 The One Dimensional Cases.- 4.1 The Continuous Case.- 4.2 The Lattice Case.- 5 Problems.- 6 Notes and Complements.- VII Absolutely Continuous Spectrum and Inverse Theory.- 1 The w-function.- 1.1 More on Herglotz Functions.- 1.2 The Continuous Case.- 1.3 The Lattice Case.- 2 Periodic and Almost Periodic Potentials.- 2.1 Floquet Theory.- 2.2 Inverse Spectral Theory.- 2.3 The Lattice Case.- 2.4 Almost Periodic Potentials.- 3 The Absolutely Continuous Spectrum.- 3.1 The Essential Support of the Absolutely Continuous Spectrum.- 3.2 Support Theorems and Deterministic Potentials.- 4 Inverse Spectral Theory.- 4.1 The Continuous Case.- 4.2 The Lattice Case.- 5 Miscellaneous.- 5.1 Potentials Taking Finitely Many Values.- 5.2 A Remark on the Multidimensional Case.- 6 Problems.- 7 Notes and Complements.- VIII Localization in One Dimension.- 1 Pointwise Theory.- 1.1 Kotani's Trick.- 1.2 The Discrete Case.- 1.3 The General Case.- 2 Perturbation Theory.- 3 Operator Theory.- 3.1 The Discrete I.I.D. Model.- 3.2 The Markov Model.- 3.3 The Discrete I.I.D. Model on the Strip.- 4 Localization for Singular Potentials.- 5 Non-Stationary Processes.- 5.1 The Discrete Case.- 5.2 The Continuous Case.- 6 Problems.- 7 Notes and Complements.- IX Localization in Any Dimension.- 1 Exponential Decay of the Green's Function at Fixed Energy.- 1.1 Decay of the Green's Function in Boxes.- 1.2 Decay of the Green's Function in ?d.- 2 Localization for A.C. Potentials.- 2.1 Pointwise Theory.- 2.2 Perturbation Theory.- 3 A Direct Proof of Localization.- 3.1 Examples.- 3.2 The Proof.- 3.3 Extensions.- 4 Problems.- 5 Notes and Complements.- Notation Index.

Spectral Theory of Self-Adjoint Operators

Abstract: The present chapter is devoted to the introduction of the notation, the definitions and most of the results from functional analysis which will be needed in the sequel. Because of lack of space, we refrain from explaining the motivations behind the numerous definitions we introduce. We merely illustrate them with examples of Schrodinger operators and we postpone a more detailed study to Chapter II. Rather than a serious introduction to the spectral theory of self-adjoint operators, this chapter should be understood as a glossary and a summary of the terms and results to be used in the sequel.

The Integrated Density of States

Abstract: This chapter is entirely devoted to the study of the so-called integrated density of states. It is an object of interest for various reasons. It is of physical importance for it can be measured experimentally in some cases. On the top of its physical appeal, the integrated density of states is a very interesting mathematical object which deserves to be investigated for its own sake. Many challenging mathematical problems remain open in this respect. Finally, several proofs of technical results crucial to the study of the spectral properties of the random Hamiltonians, and in particular of the localization, rely very heavily on estimates of the integrated density of states.

Products of Random Matrices

Abstract: Part of the one dimensional or quasi-one dimensional theory of localization can be reduced to the study of products of random matrices One of the most important result in this direction is the extension to matrix valued random variables of the strong law of large numbers Unfortunately the identification of the limit (called the Lyapunov exponent) is more complicated than in the classical case of real valued random variables In particular this limit can no longer be written as a single expectation Moreover its determination involves the computation of some invariant measure on the projective space We only assume that the reader has a minimal background in classical probability theory Most of the material presented in this chapter is self contained

Localization in One Dimension

Abstract: It was first claimed by P.W. Anderson (1958) in [8] for the multidimensional random Schrodinger operator on a lattice, (associated with i.i.d. potentials) that the spectrum ought to be pure point with exponentially decaying eigenfunctions for a “typical sample” and for “large disorder”. It was later conjectured by Mott & Twose in [249] that this property should hold in the one dimensional case at any disorder. This chapter is devoted to the proof of this last conjecture which we will extend to quasi-one dimensional systems.