Distribution of eigenvalues for some sets of random matrices

TL;DR: In this article, the authors studied the distribution of eigenvalues for two sets of random Hermitian matrices and one set of random unitary matrices in the energy spectra of disordered systems.

Abstract: In this paper we study the distribution of eigenvalues for two sets of random Hermitian matrices and one set of random unitary matrices. The statement of the problem as well as its method of investigation go back originally to the work of Dyson [i] and I. M. Lifsic [2], [3] on the energy spectra of disordered systems, although in their probability character our sets are more similar to sets studied by Wigner [4]. Since the approaches to the sets we consider are the same, we present in detail only the most typical case. The corresponding results for the other two cases are presented without proof in the last section of the paper. §1. Statement of the problem and survey of results We shall consider as acting in iV-dimensiona l unitary space ///v, a selfadjoint operator BN (re) of the form