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."

Jean Bourgain is Professor of Mathematics at the Institute for Advanced Study and J. Doob Professor of Mathematics at the University of Illinois, Urbana-Champaign. He is the author of Global Solutions of Nonlinear Schrödinger Equations.

Acknowledgment v
CHAPTER 1: Introduction 1
CHAPTER 2: Transfer Matrix and Lyapounov Exponent 11
CHAPTER 3: Herman's Subharmonicity Method 15
CHAPTER 4: Estimates on Subharmonic Functions 19
CHAPTER 5: LDT for Shift Model 25
CHAPTER 6: Avalanche Principle in SL₂( R ) 29
CHAPTER 7: Consequences for Lyapounov Exponent, IDS, and Green's Function 31
CHAPTER 8: Refinements 39
CHAPTER 9: Some Facts about Semialgebraic Sets 49
CHAPTER 10: Localization 55
CHAPTER 11: Generalization to Certain Long-Range Models 65
CHAPTER 12: Lyapounov Exponent and Spectrum 75
CHAPTER 13: Point Spectrum in Multifrequency Models at Small Disorder 87
CHAPTER 14: A Matrix-Valued Cartan-Type Theorem 97
CHAPTER 15: Application to Jacobi Matrices Associated with Skew Shifts 105
CHAPTER 16: Application to the Kicked Rotor Problem 117
CHAPTER 17: Quasi-Periodic Localization on the Z ^d -lattice ( d > 1) 123
CHAPTER 18: An Approach to Melnikov's Theorem on Persistency of Non-resonant Lower Dimension Tori 133
CHAPTER 19: Application to the Construction of Quasi-Periodic Solutions of Nonlinear Schrödinger Equations 143
CHAPTER 20: Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations 159
Appendix 169