Random Operators: Disorder Effects on Quantum Spectra and Dynamics
About this Title
Michael Aizenman, Princeton University, Princeton, NJ and Simone Warzel, Technische Universität München, München, Germany
Publication: Graduate Studies in Mathematics
Publication Year: 2015; Volume 168
ISBNs: 978-1-4704-1913-4 (print); 978-1-4704-2783-2 (online)
MathSciNet review: MR3364516
MSC: Primary 82B44; Secondary 46N50, 47B80, 60H25, 81Q10, 81Q12, 82B10, 82D30
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
Graduate students and researchers interested in random operator theory.
Table of Contents
- Chapter 1. Introduction
- Chapter 2. General relations between spectra and dynamics
- Chapter 3. Ergodic operators and their self-averaging 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 Green-Kubo-Streda 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. Herglotz-Pick functions and their spectra