Caught by Disorder - Peter Stollmann

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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 introducing readers to disorder in a hands-on way....

Sommaire:
1 Getting Started.- Prologue.- 1.1 Bound states versus extended states.- 1.2 Ergodic operator families.- 1.3 Some important examples.- 1.4 Our basic models (P+A) and (DIV).- 1.5 Localization and Lifshitz tails: the heuristic picture.- 2 Analysis of Anderson-type Models.- Prologue.- 2.1 Lifshitz tails for (P+A).- 2.2 Initial length scale estimates.- 2.3 Wegner estimates.- 2.4 Combes-Thomas estimates.- 2.5 Changing cubes.- 3 Multiscale Analysis.- Prologue.- 3.1 Idea of the proof and historical notes.- 3.2 Multiscale Analysis.- 3.3 Exponential localization.- 3.4 Dynamical localization.- 3.5 More models.- 4 Appendix.- 4.1 A short story of selfadjoint operators.- 4.2 Some basics from probability theory.- 5 Aftermath.- References.- Author Index.

The main purpose of this book is to present, in quite an accessible way, the essence of multiscale analysis, a technique needed in proving Anderson localization, or exponential localization, for random Schrodinger-like operators acting in dollarsL^2(\bold R^d) dollars. The treatise consists of four chapters, which are well arranged so as to clarify the logical structure of this hard technique. In the first chapter, after a brief introduction to the subject of disordered systems, the author summarizes some general facts on ergodic families of self-adjoint operators, such as the almost sure constancy of the spectrum. A convenient criterion is also given for the measurability of random operators obtained through closed forms. Then the author describes precisely two basic models to be treated in the sequel, which he names (P+A) and (DIV) respectively.... (Nariyuki Minami, Mathematical Reviews)