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

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." This book presents an overview of recent developments in 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, particularly 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.' Ch. 1. Introduction -- Ch. 2. Transfer Matrix And Lyapounov Exponent -- Ch. 3. Herman's Subharmonicity Method -- Ch. 4. Estimates On Subharmonic Functions -- Ch. 5. Ldt For Shift Model -- Ch. 6. Avalanche Principle In Sl[subscript 2](r) -- Ch. 7. Consequences For Lyapounov Exponent, Ids, And Green's Function -- Ch. 8. Refinements -- Ch. 9. Some Facts About Semialgebraic Sets -- Ch. 10. Localization -- Ch. 11. Generalization To Certain Long-range Models -- Ch. 12. Lyapounov Exponent And Spectrum -- Ch. 13. Point Spectrum In Multifrequency Models At Small Disorder -- Ch. 14. Matrix-valued Cartan-type Theorem -- Ch. 15. Application To Jacobi Matrices Associated With Skew Shifts -- Ch. 16. Application To The Kicked Rotor Problem -- Ch. 17. Quasi-periodic Localization On The Z[superscript D]-lattice (d > 1) -- Ch. 18. Approach To Melnikov's Theorem On Persistency Of Non-resonant Lower Dimension Tori -- Ch. 19. Application To The Construction Of Quasi-periodic Solutions Of Nonlinear Schrodinger Equations -- Ch. 20. Construction Of Quasi-periodic Solutions Of Nonlinear Wave Equations. J. Bourgain. Includes Bibliographical References. We will consider infinite matrices indexed by Z (or Zb) associated to a dynamical system in the sense that satisfies where, and T is an ergodic measure-preserving transformation of.