Random Schrödinger Operators
This is a book on a subject intensively developed over the last 50 years by theoretical physicists and also, in recent decades, by researchers stressing a full mathematical rigour in such a study, several prominent mathematical physicists among them. A lot of deep results have been accumulated over the last 30 years (and many problems remain to be rigorously solved). This book aims to give a self-contained introduction and review of the subject. There are three articles in the book: “An invitation to random Schrödinger operators” (by W. Kirsch, with an appendix on the Aizenman-Molchanov method written by F. Klopp); “Multiscale Analysis and Localization of random operators” (by A. Klein); and “Random matrices and the Anderson Model” ( by M. Disertori and V. Rivasseau).
The first article is an introductory course on the subject for the case of the discrete Laplacian. It covers all the basic aspects of the theory. The next article explains the method of multiscale analysis (which is also applicable to other classes of random Hamiltonians, like waves in random media) in the continuum. The third article describes the approach of constructive quantum theory when applied to the subject of random Schrödinger operators. It concentrates on the delocalized regime and also uses (in contrast to previous sections, which mostly have an operator theoretic or probabilistic character) tools like Grassman variables, superintegrals and other methods of field theory and random matrix theory. The authors have been successful in combining the choice of the essential themes of this theory with an accessible presentation of its basic methods, often adding remarks enlightening the motivation, the history of the subject and relations of methods used here to other parts of mathematical physics. The book will serve as a useful, self-contained source of information on this important chapter of contemporary mathematical physics.