Green's Function Estimates for Lattice Schrodinger Operators and Applications. (AM-158)
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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." G. Teschl - Monatshefte fur Mathematik fur Mathematik This text is an up to date introduction to localization problems for lattice Schrödinger operations with deterministic ergodic potentials by one of the leading experts. . . . I can recommend it to any graduate student or researcher in the field.
Ch. 1Introduction1Ch. 2Transfer matrix and Lyapounov exponent11Ch. 3Herman's subharmonicity method15Ch. 4Estimates on subharmonic functions19Ch. 5LDT for shift model25Ch. 6Avalanche principle in SL[subscript 2](R)29Ch. 7Consequences for Lyapounov exponent, IDS, and Green's function31Ch. 8Refinements39Ch. 9Some facts about semialgebraic sets49Ch. 10Localization55Ch. 11Generalization to certain long-range models65Ch. 12Lyapounov exponent and spectrum75Ch. 13Point spectrum in multifrequency models at small disorder87Ch. 14A matrix-valued cartan-type theorem97Ch. 15Application to Jacobi matrices associated with skew shifts105Ch. 16Application to the kicked rotor problem117Ch. 17Quasi-periodic localization on the Z[superscript d]-lattice (d > 1)123Ch. 18An approach to Melnikov's theorem on persistency of non-resonant lower dimension tori133Ch. 19Application to the construction of quasi-periodic solutions of nonlinear Schrodinger equations143Ch. 20Construction of quasi-periodic solutions of nonlinear wave equations159
