Overview

The study of the spectra and related characteristics of random and almost periodic operators of various types (Schrodinger, continuous, discrete and more general) is a lively field of research lying at the intersection of mathematical physics, spectral theory of operators and probability theory. A widespread interest in the domain and a considerable amount of mathematical activity have led to many new results and viewpoints yielding insight even into traditional questions. This book by two of the leading researchers is a systematic treatment of the fundamental problems and the large body of mathematical results known. The book also provides a number of exercises illustrating these results to guide the reader towards improvements and generalizations.

Full Product Details

Author:   Leonid Pastur ,  Alexander Figotin
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Volume:   Part 297
Weight:   1.010kg
ISBN:  

9783540506225

ISBN 10:   3540506225
Pages:   595
Publication Date:   16 December 1991
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of stock  
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Table of Contents

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 Schrodinger 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 Schrodinger 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 Schrodinger 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 Schrodinger Operator with a Nonlocal Quasiperiodic Potential.- Problems.- Appendix A: Nevanlinna Functions.- Appendix B: Distribution of Eigenvalues of Large Random Matrices.- List of Symbols.

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