One-dimensional ergodic Schrödinger operators

• Verfasst von: Damanik, David [VerfasserIn]
• Verfasst von: Fillman, Jake [VerfasserIn]
• Titel: One-dimensional ergodic Schrödinger operators
• Titelzusatz: part I: General theory
• Verf.angabe: David Damanik, Jake Fillman
• Verlagsort: Providence, Rhode Island
• Verlag: American Mathematical Society
• E-Jahr: 2022
• Jahr: [2022]
• Umfang: 1 Online-Ressource (xv, 444 Seiten)
• Gesamttitel/Reihe: Graduate studies in mathematics ; 221
• Fussnoten: Description based on publisher supplied metadata and other sources
• ISBN: 978-1-4704-7085-2
• Datenträger: Online-Ressource
• Sprache: eng
• Bibliogr. Hinweis: Erscheint auch als : Druck-Ausgabe: Damanik, David, 1971 - : One-dimensional ergodic Schrödinger operators ; 1: General theory. - Providence, Rhode Island : American Mathematical Society, 2022. - xv, 444 Seiten
• RVK-Notation: SK 620
• Sach-SW: Electronic books
• K10plus-PPN: 1811690149
• K10plus-PPN:
  Universitätsbibliothek
• Lokale URL UB: Zum Volltext

Abstract

The theory of one-dimensional ergodic operators involves a beautiful synthesis of ideas from dynamical systems, topology, and analysis. Additionally, this setting includes many models of physical interest, including those operators that model crystals, disordered media, or quasicrystals. This field has seen substantial progress in recent decades, much of which has yet to be discussed in textbooks.Beginning with a refresher on key topics in spectral theory, this volume presents the basic theory of discrete one-dimensional Schrödinger operators with dynamically defined potentials. It also includes a self-contained introduction to the relevant aspects of ergodic theory and topological dynamics.This text is accessible to graduate students who have completed one-semester courses in measure theory and complex analysis. It is intended to serve as an introduction to the field for junior researchers and beginning graduate students as well as a reference text for people already working in this area. It is well suited for self-study and contains numerous exercises (many with hints).

Abstract

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.