Caught by Disorder: Bound States in Random Media
Springer Science & Business Media, 2001 - 166 Seiten
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 multiscale 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 partial differential equations, functional analysis---mainly operator theory and spectral theory---and elementary probability theory. The work is an excellent text for a graduate course or seminar in mathematical physics or serves as a standard reference for specialists.
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absolutely continuous Anderson Localization Assume assumption band edges basic models boundary conditions bounded boxes called compact consider defined denotes disjoint disorder domain dynamical localization eigenfunction eigenvalues electrons energy ergodic exists exponential decay exponential localization fact fixed fluctuation boundaries G G G H(ai hamiltonian Hilbert space independent induction inequality initial length scale integrated density Kirsch Laplacian Lemma length scale estimates Let H Lifshitz asymptotics Lifshitz tails linear Math mathematical measurable functions Moreover multiscale analysis norm Notes and Remarks open cube operator H perturbation Phys polynomially Preprint probability measure probability space properties prove pure point spectrum random operators random potential random Schrodinger operators random variables respect restriction Riesz Representation Theorem satisfied Schrodinger equation Section selfadjoint operator selfadjoint realization sequence sesquilinear form sidelength single-site measure single-site potential Sobolev space Spectral Theorem Stollmann subspace theory tion waves Wegner estimate
Partial Differential Equations and Inverse Problems
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