Spectral Theory of Random Schroedinger Operators: A Genetic Introduction

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The interplay between the spectral theory of Schroedinger operators and probabilistic considerations forms the main theme of these notes, written for the non-specialist reader and intended to provide a brief and elementary introduction to this field. An attempt is made to show basic ideas in statu nascendi and to follow their evaluation from simple beginnings through to more advanced results. The term genetic in the title refers to this procedure. The author concentrates on two topics which, in the history of the subject, have been of major conceptual importance. On the one hand, the Laplacian is a random medium and the left end of its spectrum (leading to large deviation problems for Brownian motion and the link to the notion of entropy) and on the other, Schroedinger operators with general ergodic potentials in one-dimensional space. Ideas and concepts are explained in the simple terms and by means of a few characteristic problems with heuristic arguments preceding rigorous proofs.