Volume: 168; 2015; 326 pp; Hardcover
MSC: Primary 82; 60; 47; 81; 46;
Print ISBN: 978-1-4704-1913-4
Product Code: GSM/168
List Price: $84.00
AMS Member Price: $67.20
MAA Member Price: $75.60
Electronic ISBN: 978-1-4704-2783-2
Product Code: GSM/168.E
List Price: $79.00
AMS Member Price: $63.20
MAA Member Price: $71.10
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Supplemental Materials
Random Operators: Disorder Effects on Quantum Spectra and Dynamics
Share this pageMichael Aizenman; Simone Warzel
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 self-adjoint
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 Herglotz-Pick 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
Readership
Graduate students and researchers interested in random operator theory.
Table of Contents
Table of Contents
Random Operators: Disorder Effects on Quantum Spectra and Dynamics
- Cover Cover11
- Title page iii4
- Dedication v6
- Contents vii8
- Preface xiii14
- Chapter 1. Introduction 116
- Chapter 2. General relations between spectra and dynamics 1126
- Chapter 3. Ergodic operators and their self-averaging properties 2742
- Chapter 4. Density of states bounds: Wegner estimate and Lifshitz tails 4560
- Chapter 5. The relation of Green functions to eigenfunctions 6984
- Chapter 6. Anderson localization through path expansions 91106
- Chapter 7. Dynamical localization and fractional moment criteria 101116
- Chapter 8. Fractional moments from an analytical perspective 117132
- Chapter 9. Strategies for mapping exponential decay 135150
- Chapter 10. Localization at high disorder and at extreme energies 149164
- Chapter 11. Constructive criteria for Anderson localization 165180
- Chapter 12. Complete localization in one dimension 175190
- Chapter 13. Diffusion hypothesis and the Green-Kubo-Streda formula 199214
- Chapter 14. Integer quantum Hall effect 215230
- Chapter 15. Resonant delocalization 233248
- Chapter 16. Phase diagrams for regular tree graphs 249264
- Chapter 17. The eigenvalue point process and a conjectured dichotomy 269284
- Appendix A. Elements of spectral theory 289304
- Appendix B. Herglotz-Pick functions and their spectra 299314
- Bibliography 303318
- Index 323338
- Back Cover Back Cover1343