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Green's Function Estimates for Lattice Schroedinger Operators and Applications. (AM-158) : Annals of Mathematics Studies - Jean Bourgain

Green's Function Estimates for Lattice Schroedinger Operators and Applications. (AM-158)

Annals of Mathematics Studies


Published: 21st November 2004
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This book presents an overview of recent developments in the area of localization for quasi-periodic lattice Schrodinger 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."

"This text is an up to date introduction to localization problems for lattice Schr?dinger operations with deterministic ergodic potentials by one of the leading experts... I can recommend it to any graduate student or researcher in the field."--G. Teschl, Monatschefte fur Mathematik

Acknowledgmentp. v
Introductionp. 1
Transfer Matrix and Lyapounov Exponentp. 11
Herman's Subharmonicity Methodp. 15
Estimates on Subharmonic Functionsp. 19
LDT for Shift Modelp. 25
Avalanche Principle in SL 2 ( R )p. 29
Consequences for Lyapounov Exponent, IDS, and Green's Functionp. 31
Refinementsp. 39
Some Facts about Semialgebraic Setsp. 49
Localizationp. 55
Generalization to Certain Long-Range Modelsp. 65
Lyapounov Exponent and Spectrump. 75
Point Spectrum in Multifrequency Models at Small Disorderp. 87
A Matrix-Valued Cartan-Type Theoremp. 97
Application to Jacobi Matrices Associated with Skew Shiftsp. 105
Application to the Kicked Rotor Problemp. 117
Quasi-Periodic Localization on the Z d -lattice ( d >1)p. 123
An Approach to Melnikov's Theorem on Persistency of Non-resonant Lower Dimension Torip. 133
Application to the Construction of Quasi-Periodic Solutions of Nonlinear Schrouml;dinger Equationsp. 143
Construction of Quasi-Periodic Solutions of Nonlinear Wave Equationsp. 159
Appendixp. 169
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691120980
ISBN-10: 0691120986
Series: Annals of Mathematics Studies
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 200
Published: 21st November 2004
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 1.91
Weight (kg): 0.31