This paper represents a very informative review of results and problems in the spectral theory of self-adjoint random operators. An important example of such an operator is the lattice Schrödinger operator \({\mathbb{H}}_ q=-\Delta_ d+q\) in \(\ell_ 2({\mathbb{Z}}^ d)\), where \(\Delta_ d\) denotes the discrete Laplacian and the q(x), \(x\in {\mathbb{Z}}^ d\), are i.i.d. random variables (Anderson’s tight-binding model). The operator \({\mathbb{H}}_ d\) belongs to the set of so-called metrically transitive operators (MTO) generally studied by the author in the first chapter of his paper. In the next section he deals with general differential or matrix operators belonging to the set of MTO’s.
Then the subset of one-dimensional second-order finite-difference and differential operators is considered. In particular, the connections between the Lyapunov exponent and the spectral theory are represented in detail. The results of this section are based on the special structure of the corresponding equations and not on special assumptions on the random fields considered beside of their metric transitivity.
In the following 4th section, the author supposes the maximum of randomness of these fields, namely that they are sequences of independent random variables (in the discrete case) or ergodic Markov processes (in the continuous case). The 5th chapter is devoted to the asymptotic behaviour of the integrated state density on the boundaries of the spectrum in the multi-dimensional case. Finally, in the last chapter multi-dimensional MTO’s are discussed. At the end of his paper the author refers to unsolved problems not considered in the main text. The paper contains 159 references, 85 of which written by Soviet authors.