Spectra of Random and Almost-Periodic Operators

ISBN-13: 9783642743481 / Angielski / Miękka / 2011 / 587 str.

Leonid Pastur; Alexander Figotin

Opis

In the last fifteen years the spectral properties of the Schrodinger equation and of other differential and finite-difference operators with random and almost-periodic coefficients have attracted considerable and ever increasing interest. This is so not only because of the subject's position at the in tersection of operator spectral theory, probability theory and mathematical physics, but also because of its importance to theoretical physics, and par ticularly to the theory of disordered condensed systems. It was the requirements of this theory that motivated the initial study of differential operators with random coefficients in the fifties and sixties, by the physicists Anderson, 1. Lifshitz and Mott; and today the same theory still exerts a strong influence on the discipline into which this study has evolved, and which will occupy us here. The theory of disordered condensed systems tries to describe, in the so-called one-particle approximation, the properties of condensed media whose atomic structure exhibits no long-range order. Examples of such media are crystals with chaotically distributed impurities, amorphous substances, biopolymers, and so on. It is natural to describe the location of atoms and other characteristics of such media probabilistically, in such a way that the characteristics of a region do not depend on the region's position, and the characteristics of regions far apart are correlated only very weakly. An appropriate model for such a medium is a homogeneous and ergodic, that is, metrically transitive, random field."

Spis treści

I. Metrically Transitive Operators.- 1 Basic Definitions and Examples.- 1.A Random Variables, Functions and Fields.- 1.B Random Vectors and Operators.- l.C Metrically Transitive Random Fields.- l.D Metrically Transitive Operators.- 2 Simple Spectral Properties of Metrically Transitive Operators.- 2.A Deficiency Indices.- 2.B Nonrandomnessofthe Spectrum and of its Components.- 2.C Nonrandomness of Multiplicities.- Problems.- II. Asymptotic Properties of Metrically Transitive Matrix and Differential Operators.- 3 Review of Basic Results.- 4 Matrix Operators on ?2 (Zd).- 4.A Essential Self-Adjointness.- 4.B Existence of the Integrated Density of States and Other Ergodic Properties.- 4.C Simple Properties of the Integrated Density of States and of the Spectra of Metrically Transitive Matrix Operators.- 4.D Location of the Spectrum.- 5 Schrödinger Operators and Elliptic Differential Operators on L2(Rd).- 5.A Criteria for Essential Self-Adjointness.- 5.B Ergodic Properties.- 5.C Some Properties of the Integrated Density of States.- 5.D Location of the Spectrum of a Metrically Transitive Schrödinger Operator.- Problems.- III. Integrated Density of States in One-Dimensional Problems of Second Order.- 6 The Oscillation Theorem and the Integrated Density of States.- 6. A The Phase and the Existence of the Integrated Density of States.- 6.B Simplest Asymptotics of the Integrated Density of States at the Edges of the Spectrum.- 6.C Schrödinger Operator with Markov Potential.- 6.D The Brownian Motion Model.- 6.E Jacobi Matrices with Independent and Markov Coefficients.- 6.F Smoothness of N (?); Special Energies.- 7 Examples of Calculation of the Integrated Density of States.- 7.A The Kronig-Penny Stochastic Model.- 7.B Random Jacobi Matrices.- Problems.- IV. Asymptotic Behavior of the Integrated Density of States at Spectral Boundaries in Multidimensional Problems.- 8 Stable Boundaries.- 9 Fluctuation Boundaries: General Discussion and Classical Asymptotics.- 9.A Introduction and Heuristic Discussion.- 9.B Simplest Bounds. Gaussian and Negative Poisson Potentials.- 9.C Generalized Poisson Potential.- 10 Fluctuation Boundaries: Quantum Asymptotics.- 10.A The Lifshitz Exponent.- 10.B Generalized Poisson Potential with a Nonnegative, Rapidly Decreasing Function.- 10.C Smoothed Square of a Gaussian Random Field.- Problems.- V. Lyapunov Exponents and the Spectrum in One Dimension.- 11 Existence and Properties of Lyapunov Exponents.- 11.A The Multiplicative Ergodic Theorem and the Existence of Lyapunov Exponents.- 11.B The Lyapunov Exponent and the Integrated Density of States.- 11.C Simplest Asymptotic Formulas and Estimates for Lyapunov Exponents.- 12 Lyapunov Exponents and the Absolutely Continuous Spectrum.- 12.A Basic Facts About the Spectrum of One-Dimensional Operators of the Second Order.- 12.B Lyapunov Exponents and the Absolutely Continuous Spectrum.- 12.C Multiplicity of the Spectrum.- 12.D Deterministic Potentials.- 12.E Some Inverse Problems.- 13 Lyapunov Exponents and the Point Spectrum.- 13.A Heuristic Discussion.- 13.B Conditions for Positive Lyapunov Exponents to Imply a Pure Point Spectrum.- Problems.- VI. Random Operators.- 14 The Lyapunov Exponent of Random Operators in One Dimension.- 14.A Positiveness of the Lyapunov Exponent.- 14.B Asymptotic Formulas for the Lyapunov Exponent.- 15 The Point Spectrum of Random Operators.- 15.A The Pure Point Spectrum in One Dimension.- 15.B Other One-Dimensional Results.- 15.C The Point Spectrum in Multidimensional Problems.- Problems.- VII. Almost-Periodic Operators.- 16 Smooth Quasi-Periodic Potentials.- 16.A The Integrated Density of States and the Gap Labeling Theorem.- 16.B Absolutely Continuous Spectrum.- 16.C Lower Bounds of Solutions and Absence of a Point Spectrum.- 16.D Lower Bounds for the Lyapunov Exponent and Absence of an Absolutely Continuous Spectrum in the Discrete Case.- 16.E Point Spectrum of Almost-Periodic Operators.- 16.F The Almost-Mathieu Operator.- 17 Limit-Periodic Potentials.- 17.A Basic Results.- 17.B Spectral Data for Periodic Potentials of Increasing Period.- 17.C Proof of the Main Theorems.- 18 Unbounded Quasiperiodic Potentials.- 18.A General Results and the Integrated Density of States.- 18.B The Case of Strongly Incommensurate Frequencies.- 18.C The One-Dimensional Case.- 18.D The Schrödinger Operator with a Nonlocal Quasiperiodic Potential.- Problems.- Appendix A: Nevanlinna Functions.- Appendix B: Distribution of Eigenvalues of Large Random Matrices.- List of Symbols.