Green's function estimates for lattice Schrödinger operators and applications

Author(s)

• Bourgain, Jean

Bibliographic Information

Green's function estimates for lattice Schrödinger operators and applications

J. Bourgain

（Annals of mathematics studies, no. 158）

Princeton University Press, c2005

• : hbk
• : pbk

Note

Includes bibliographical references

Description and Table of Contents

Volume

: hbk ISBN 9780691120973

Description

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, this 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

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 SL<SUB>2( 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 Schrodinger Equations 143 CHAPTER 20: Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations 159 Appendix 169

Volume

: pbk ISBN 9780691120980

Description

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."

Table of Contents

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 Schrodinger Equations 143 CHAPTER 20: Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations 159 Appendix 169

by "Nielsen BookData"

• NCID: BA69960059
• ISBN:
  • 0691120978
  • 0691120986
• LCCN: 2004104492
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: Princeton
• Pages/Volumes: vi, 173 p.
• Size: 24 cm
• Classification:
  • LCC : QC174.17.S3
• Subject Headings:
  • LCSH : Schrödinger operators
  • LCSH : Green's functions
  • LCSH : Hamiltonian systems
  • LCSH : Evolution equations
• Parent Bibliography ID:
  • BA00300336