Green's Function Estimates for Lattice Schrödinger Operators and Applications

Jean Bourgain

• Publisher: Princeton University Press
• Year: 2004
• Print ISBN:
  9780691120980
  9780691120973
• eBook ISBN:
  9781400837144
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

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."

Table of contents

• Cover
• Title
• Copyright
• 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 LyapounovExponent, 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 Z^d-lattice (d > 1)
• Chapter 18. An Approach to Melnikov’s Theorem on Persistency of Non-resonant 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