Spectra of random self adjoint operators

This survey contains an exposition of the results obtained in the studying the spectra of certain classes of random operators. It consists of three chapters. In the introductory Chapter I we survey some of the pioneering papers (two, in particular), which have sufficient depth of content to suggest the natural problems to be considered in this field. In Chapter II we study the distribution of the eigenvalues for ensembles of random matrices, for instance, the sum of one-dimensional projection operators onto random vectors uniformly and independently distributed over the surface of the $n$-dimensional unit sphere. We show that as $n\to\infty$, the eigenvalue distribution ceases to be random and can be determined as the solution of a certain functional equation. Chapter III deals with the Schrödinger equation with a random potential. We establish ergodic properties of certain random quantities, constructed from the eigenvalues and eigenfunctions of this equation, and we study the distribution of eigenvalues in the cases when the potential is a Gaussian random field and a homogeneous Markov process.