Hardcover ISBN:  9781470419134 
Product Code:  GSM/168 
326 pp 
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AMS Member Price:  $67.20 
Electronic ISBN:  9781470427832 
Product Code:  GSM/168.E 
326 pp 
List Price:  $79.00 
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Book DetailsGraduate Studies in MathematicsVolume: 168; 2015MSC: Primary 82; 60; 47; 81; 46;
This book provides an introduction to the mathematical theory of disorder effects on quantum spectra and dynamics. Topics covered range from the basic theory of spectra and dynamics of selfadjoint operators through Anderson localization—presented here via the fractional moment method, up to recent results on resonant delocalization.
The subject's multifaceted presentation is organized into seventeen chapters, each focused on either a specific mathematical topic or on a demonstration of the theory's relevance to physics, e.g., its implications for the quantum Hall effect. The mathematical chapters include general relations of quantum spectra and dynamics, ergodicity and its implications, methods for establishing spectral and dynamical localization regimes, applications and properties of the Green function, its relation to the eigenfunction correlator, fractional moments of HerglotzPick functions, the phase diagram for tree graph operators, resonant delocalization, the spectral statistics conjecture, and related results.
The text incorporates notes from courses that were presented at the authors' respective institutions and attended by graduate students and postdoctoral researchers.It has been almost 25 years since the last major book on this subject. The authors masterfully update the subject but more importantly present their own probabilistic insights in clear fashion. This wonderful book is ideal for both researchers and advanced students.
—Barry Simon, California Institute of Technology
ReadershipGraduate students and researchers interested in random operator theory.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. General relations between spectra and dynamics

Chapter 3. Ergodic operators and their selfaveraging properties

Chapter 4. Density of states bounds: Wegner estimate and Lifshitz tails

Chapter 5. The relation of Green functions to eigenfunctions

Chapter 6. Anderson localization through path expansions

Chapter 7. Dynamical localization and fractional moment criteria

Chapter 8. Fractional moments from an analytical perspective

Chapter 9. Strategies for mapping exponential decay

Chapter 10. Localization at high disorder and at extreme energies

Chapter 11. Constructive criteria for Anderson localization

Chapter 12. Complete localization in one dimension

Chapter 13. Diffusion hypothesis and the GreenKuboStreda formula

Chapter 14. Integer quantum Hall effect

Chapter 15. Resonant delocalization

Chapter 16. Phase diagrams for regular tree graphs

Chapter 17. The eigenvalue point process and a conjectured dichotomy

Appendix A. Elements of spectral theory

Appendix B. HerglotzPick functions and their spectra


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 Book Details
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 Additional Material

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This book provides an introduction to the mathematical theory of disorder effects on quantum spectra and dynamics. Topics covered range from the basic theory of spectra and dynamics of selfadjoint operators through Anderson localization—presented here via the fractional moment method, up to recent results on resonant delocalization.
The subject's multifaceted presentation is organized into seventeen chapters, each focused on either a specific mathematical topic or on a demonstration of the theory's relevance to physics, e.g., its implications for the quantum Hall effect. The mathematical chapters include general relations of quantum spectra and dynamics, ergodicity and its implications, methods for establishing spectral and dynamical localization regimes, applications and properties of the Green function, its relation to the eigenfunction correlator, fractional moments of HerglotzPick functions, the phase diagram for tree graph operators, resonant delocalization, the spectral statistics conjecture, and related results.
The text incorporates notes from courses that were presented at the authors' respective institutions and attended by graduate students and postdoctoral researchers.
It has been almost 25 years since the last major book on this subject. The authors masterfully update the subject but more importantly present their own probabilistic insights in clear fashion. This wonderful book is ideal for both researchers and advanced students.
—Barry Simon, California Institute of Technology
Graduate students and researchers interested in random operator theory.

Chapters

Chapter 1. Introduction

Chapter 2. General relations between spectra and dynamics

Chapter 3. Ergodic operators and their selfaveraging properties

Chapter 4. Density of states bounds: Wegner estimate and Lifshitz tails

Chapter 5. The relation of Green functions to eigenfunctions

Chapter 6. Anderson localization through path expansions

Chapter 7. Dynamical localization and fractional moment criteria

Chapter 8. Fractional moments from an analytical perspective

Chapter 9. Strategies for mapping exponential decay

Chapter 10. Localization at high disorder and at extreme energies

Chapter 11. Constructive criteria for Anderson localization

Chapter 12. Complete localization in one dimension

Chapter 13. Diffusion hypothesis and the GreenKuboStreda formula

Chapter 14. Integer quantum Hall effect

Chapter 15. Resonant delocalization

Chapter 16. Phase diagrams for regular tree graphs

Chapter 17. The eigenvalue point process and a conjectured dichotomy

Appendix A. Elements of spectral theory

Appendix B. HerglotzPick functions and their spectra