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One-dimensional ergodic Schrödinger operators /

Bibliographic Details
Main Authors: Damanik, David, 1971- (Author), Fillman, Jake, 1987- (Author)
Corporate Author: EBSCOhost
Format: eBook
Language:English
Published: Providence, Rhode Island : American Mathematical Society, [2022]
Series:Graduate studies in mathematics, volume 221
Subjects:
Schrödinger operator.
Ergodic theory.
Differentiable dynamical systems.
Electronic books.
Online Access:Connect to the full text of this electronic book
Table of Contents:
  • Cover
  • Title page
  • Contents
  • Preface
  • Part I: General Theory
  • Chapter 1. Snippets from Spectral Theory
  • 1.1. Introduction
  • 1.2. A Crash Course on Banach and Hilbert Spaces
  • 1.3. Bounded Linear Operators
  • 1.4. Self-Adjoint Operators and the Spectral Theorem
  • 1.5. Dual Spaces and Locally Convex Topologies
  • 1.6. Spectral Decompositions and Quantum Dynamics
  • 1.7. Uniform and Semi-Uniform Localization Properties
  • 1.8. Refined Spectral Decompositions and Transport Exponents
  • 1.9. Stieltjes Transforms and Derivatives of Finite Measures
  • 1.10. Poltoratski's Theorem
  • 1.11. Poltoratski-Remling's Theorem
  • 1.12. Measurability of Eigenfunctions à la Gordon-Kechris
  • 1.13. SL(2, R) and SL(2, C)
  • 1.14. Avalanche Principle
  • 1.15. Further Reading and Miscellanea
  • Chapter 2. Schrödinger Operators in ℓ²( Z)
  • 2.1. Introduction
  • 2.2. Definitions and Basic Identities
  • 2.3. Oscillation Theory
  • 2.4. Spectrum and Generalized Eigenfunctions
  • 2.5. The Combes-Thomas Estimate
  • 2.6. Basic Bounds on Spreading
  • 2.7. The Jitomirskaya-Last Inequality and Subordinacy Theory
  • 2.8. Fractional Subordinacy and Refined Spectral Decompositions
  • 2.9. The Last-Simon Description of the Absolutely Continuous Part
  • 2.10. Quantum Dynamics à la Damanik and Tcheremchantsev
  • 2.11. Further Reading and Miscellanea
  • Chapter 3. Snippets from Ergodic Theory and Topological Dynamics
  • 3.1. Introduction
  • 3.2. Definitions and Examples
  • 3.3. The Birkhoff Pointwise Ergodic Theorem
  • 3.4. Kingman's Subadditive Ergodic Theorem
  • 3.5. Ergodicity in the Topological Setting
  • 3.6. Minimality
  • 3.7. Topological Entropy
  • 3.8. Unimodular Cocycles
  • 3.9. Flows, Suspensions, and the Schwartzman Homomorphism
  • 3.10. Further Reading and Miscellanea
  • Chapter 4. General Results for Ergodic Schrödinger Operators
  • 4.1. Introduction
  • 4.2. Nonrandomness of Spectra and Spectral Types
  • 4.3. The Integrated Density of States
  • 4.4. The Lyapunov Exponent
  • 4.5. Subharmonicity of the Lyapunov Exponent
  • 4.6. The Thouless Formula and Applications
  • 4.7. Kotani Theory in the Measurable Setting
  • 4.8. Kotani Theory in the Topological Setting
  • 4.9. Uniform Results in the Topological Setting
  • 4.10. The Gap-Labelling Theorem
  • 4.11. An Invitation to Ergodic Jacobi Matrices
  • 4.12. Further Reading and Miscellanea
  • Appendix A. Tools from Harmonic Analysis
  • A.1. The Fourier and Hilbert Transforms
  • A.2. Potential Theory
  • A.3. Subharmonic Functions
  • Glossary of Notation
  • Bibliography
  • Index
  • Back Cover