A Class of Matrix Ensembles

A class of random matrix ensembles is defined, with the purpose of providing a realistic statistical description of the Hamiltonian of a complicated quantum‐mechanical system (such as a heavy nucleus) for which an approximate model Hamiltonian is known. An ensemble of the class is specified by the model Hamiltonian H0, an observed eigenvalue distribution‐function r(E), and a parameter τ which may be considered to be a fictitious ``time.'' Each of H0, r(E), and τ may be chosen independently. The ensemble consists of matrices M which are obtained from H0 by an invariant random Brownian‐motion process, lasting for a time τ and tending to pull the eigenvalues of M toward the distribution r(E). For small τ the ensemble allows only small perturbations of H0. As τ → ∞, the ensemble tends to a stationary limit independent of H0 and depending on r(E) alone. The following quantitative results are obtained. (1) It is proved that the global eigenvalue distribution in the limit τ → ∞ becomes identical with the observe...