
Résumé
Disorder is one of the central topics in science today.
Over the past 15 years various aspects of the effects of
disorder have changed a number of paradigms in mathematics
and physics. One such effect is a phenomenon called
localization, which describes the very strange behavior of
waves in random media. Instead of traveling through space
as they do in ordered environments, localized waves stay in
a confined region and are caught by disorder. This work is
the first treatment of the subject in monograph or textbook
form.
The study of disorder has generated enormous research
activity in mathematics and physics. Over the past 15 years
various aspects of the subject have changed a number of
paradigms and have inspired the discovery of deep
mathematical techniques to deal with complex problems
arising from the effects of disorder. One important effect
is a phenomenon called localization, which describes the
very strange behavior of waves in random media – the
fact that waves, instead of traveling through space as they
do in ordered environments, stay in a confined region
(caught by disorder). To date, there is no treatment of
this subject in monograph or textbook form. This book fills
that gap.
'Caught by Disorder' presents:
- an introduction to disorder that can be grasped by graduate students in a hands-on way
- a concise, mathematically rigorous examination of some particular models of disordered systems
- a detailed application of the localization phenomenon, worked out in two typical model classes that keep the technicalities at a reasonable level
- a thorough examination of new mathematical machinery, in particular, the method of multi-scale analysis
- a number of key unsolved problems
- an appendix containing the prerequisites of operator theory, as well as other proofs
- examples, illustrations, comprehensive
bibliography, author and keyword index.
Mathematical background for this book requires only a knowledge of pdes, functional analysis – mainly operator theory and spectral theory – and elementary probability theory. The work will be an excellent text for a graduate course or seminar in mathematical physics or will serve as a standard reference for specialists.
Table of Contents:
Introduction
1. Bound States versus Extended States
2. Ergodic Operator Families: Definition and Abstract
Theory
3. Some Important Examples
4. Our Basic Models (P + A) and (DIV)
5. Localization and Lifshitz Tails: The Heuristic
Picture
6. Lifshitz Tails for (P + A)
7. Initial Length Tail Estimates
8. Wegener Estimates
9. Combes--Thomas Estimates
10. Changing Cubes
11. Idea of the Proof of Localization and Historical
Notes
12. Multi-scale Analysis
13. Exponential Localization
14. Dynamical Localization
15. More Models
16. Appendix: A Short Story of Selfadjoint Operators
17. Aftermath
Bibliography
Index
Caractéristiques techniques
PAPIER | |
Éditeur(s) | Birkhäuser |
Auteur(s) | Peter Stollmann |
Parution | 04/12/2001 |
Nb. de pages | 166 |
Format | 16 x 24 |
Couverture | Relié |
Poids | 341g |
Intérieur | Noir et Blanc |
EAN13 | 9780817642105 |
ISBN13 | 978-0-8176-4210-5 |
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