Green's function estimates for lattice Schrödinger operators and applications / J. Bourgain.

Format:

Book

Author/Creator:

Bourgain, Jean, 1954-2018, author.

Series:

Annals of mathematics studies ; Number 158.

Annals of Mathematics Studies ; Number 158

Language:

English

Subjects (All):

Schrödinger operator.

Green's functions.

Hamiltonian systems.

Evolution equations.

Physical Description:

1 online resource (184 p.)

Edition:

1st ed.

Place of Publication:

Princeton, New Jersey : Princeton University Press, 2005.

Language Note:

English

Summary:

This book presents an overview of recent developments in the area of localization for quasi-periodic lattice Schrödinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. The physical motivation of these models extends back to the works of Rudolph Peierls and Douglas R. Hofstadter, and the models themselves have been a focus of mathematical research for two decades. Jean Bourgain here sets forth the results and techniques that have been discovered in the last few years. He puts special emphasis on so-called "non-perturbative" methods and the important role of subharmonic function theory and semi-algebraic set methods. He describes various applications to the theory of differential equations and dynamical systems, in particular to the quantum kicked rotor and KAM theory for nonlinear Hamiltonian evolution equations. Intended primarily for graduate students and researchers in the general area of dynamical systems and mathematical physics, the book provides a coherent account of a large body of work that is presently scattered in the literature. It does so in a refreshingly contained manner that seeks to convey the present technological "state of the art."

Contents:

Front matter

Contents

Acknowledgment

Chapter 1. Introduction

Chapter 2. Transfer Matrix and Lyapounov Exponent

Chapter 3. Herman's Subharmonicity Method

Chapter 4. Estimates on Subharmonic Functions

Chapter 5. LDT for Shift Model

Chapter 6. Avalanche Principle in SL2(R)

Chapter 7. Consequences for Lyapounov Exponent, IDS, and Green's Function

Chapter 8. Refinements

Chapter 9. Some Facts about Semialgebraic Sets

Chapter 10. Localization

Chapter 11. Generalization to Certain Long-Range Models

Chapter 12. Lyapounov Exponent and Spectrum

Chapter 13. Point Spectrum in Multifrequency Models at Small Disorder

Chapter 14. A Matrix-Valued Cartan-Type Theorem

Chapter 15. Application to Jacobi Matrices Associated with Skew Shifts

Chapter 16. Application to the Kicked Rotor Problem

Chapter 17. Quasi-Periodic Localization on the Zd-lattice (d > 1)

Chapter 18. An Approach to Melnikov's Theorem on Persistency of Nonresonant Lower Dimension Tori

Chapter 19. Application to the Construction of Quasi-Periodic Solutions of Nonlinear Schrödinger Equations

Chapter 20. Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations

Appendix

Notes:

Description based upon print version of record.

Includes bibliographical references at the end of each chapters.

Description based on print version record.

ISBN:

9781400837144

1400837146

9780691120980

0691120986

OCLC:

1004872417