Caught by disorder. Bound states in random media. (English) Zbl 0983.82016

Stollmann, Peter

Progress in Mathematical Physics. 20. Boston: Birkhäuser. xvi, 166 p. (2001).

This concise monograph is devoted to the mathematical study of wave propagation in disordered media, with special emphasis on the phenomenon of localization. Technically that involves spectral analysis of various random operators (like the Schrödinger operator with random potential, or partial differential operators with random coefficients).
A mathematically precise definition of localization amounts to the proof that the underlying model operator exhibits dense pure point spectrum almost surely. The whole machinery of Wegner, Combes-Thomas estimates and multiscale analysis is employed to elucidate various aspects of localization: exponential dynamical, spectral, at weak or large disorder.
For two major classes of models a proof of exponential localization is given. Namely, in certain energy intervals, almost all operators in the pertinent random family, exhibit pure point spectrum with exponentially decaying eigenfunctions.

Reviewer: Piotr Garbaczewski (Zielona Gora)

MSC:

82C44	Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
82-02	Research exposition (monographs, survey articles) pertaining to statistical mechanics
60H25	Random operators and equations (aspects of stochastic analysis)
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis
47B80	Random linear operators

Keywords:

discorder; localization; random Schrödinger operators; Anderson models; Lifschitz tails; multiscale analysis