Table of contents

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 -dimensional unit sphere. We show that as , 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.

Chapter I establishes the essential properties of the -matrix of a passive multipole depending on the number of its branches. These properties are based on Langevin's theorem. A classification of the basic objects of investigation: -expanding matrix-functions (class ), and also positive matrix functions (class ), is introduced. Chapter II gives an account of a theory of matrix functions of class . It also investigates the simplest (elementary and primary) matrices of this class. The fact is established that elementary (and primary) factors can be split off from a given matrix of class . In particular, the factorizability of a rational reactive matrix of class is established. Chapters III-IV set forth a theory of various subclasses of matrix functions of class : , , . The realizability of the matrix functions of each of these subclasses as -matrices of passive multipoles with the corresponding provision for branches is established. The fact that they are realizable is proved by the construction of a corresponding multipole. The last chapter is concerned with a generalization of Darlington's theorem, which leads to a realization of functions of the subclasses and as -matrices or -matrices of dissipative multipoles.

For a function holomorphic in an open set the paper solves problems on the relationships between its properties along , the boundary of , on the one hand and along , the closure of , on the other. The properties discussed are those that can be expressed in terms of the derivatives, moduli of continuity, and rates of decrease or increase of the function along and along . The results are established for very wide classes of sets and majorants of the moduli of continuity. In particular, all the main results are true for every bounded simply-connected domain and any majorant of the type of a modulus of continuity. A number of problems posed in 1942 by Sewell are solved.