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Random Operators: Disorder Effects on Quantum Spectra and Dynamics

Michael Aizenman Princeton University, Princeton, NJ
Simone Warzel Technische Universität München, München, Germany
Available Formats:
Hardcover ISBN: 978-1-4704-1913-4
Product Code: GSM/168
326 pp
List Price: $84.00 MAA Member Price:$75.60
AMS Member Price: $67.20 Electronic ISBN: 978-1-4704-2783-2 Product Code: GSM/168.E 326 pp List Price:$79.00
MAA Member Price: $71.10 AMS Member Price:$63.20
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List Price: $126.00 MAA Member Price:$113.40
AMS Member Price: $100.80 Click above image for expanded view Random Operators: Disorder Effects on Quantum Spectra and Dynamics Michael Aizenman Princeton University, Princeton, NJ Simone Warzel Technische Universität München, München, Germany Available Formats:  Hardcover ISBN: 978-1-4704-1913-4 Product Code: GSM/168 326 pp  List Price:$84.00 MAA Member Price: $75.60 AMS Member Price:$67.20
 Electronic ISBN: 978-1-4704-2783-2 Product Code: GSM/168.E 326 pp
 List Price: $79.00 MAA Member Price:$71.10 AMS Member Price: $63.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$126.00
MAA Member Price: $113.40 AMS Member Price:$100.80
• Book Details

Volume: 1682015
MSC: 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 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.

• Chapters
• 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

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Volume: 1682015
MSC: 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 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.

• Chapters
• 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
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