A Class of Matrix Ensembles

Abstract

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 observed distribution r(E). (2) A nonlinear partial differential equation is obtained for the global eigenvalue distribution ρ(E, τ) as a function of E and τ. Solution of this equation will show how the distribution changes from the initial form specified by H₀ at τ = 0 to the final form r(E) at τ = ∞. Approximate solution shows that deviations of ρ(E, τ) from r(E) extending over an interval containing m eigenvalues will disappear exponentially as soon as τ is of the order of mD², where D is the local mean level spacing. (3) Exact analytic expressions are obtained for the correlation functions representing the probabilities for finding n eigenvalues at assigned positions (E₁, …, E_n) in the ensemble in the limit τ → ∞, irrespective of the positions of the remaining (N‐n) eigenvalues. It is made plausible, but not proved, that these correlation functions tend to limits as N → ∞, which are universal functions independent of r(E). If proved, this statement would imply that the local statistical properties (spacing distributions, etc.) of eigenvalues in the ensemble become, when τ and N are both large, universal properties independent of the global eigenvalue distributions. In particular, the spacing distributions would be identical with those calculated for more special ensembles by Wigner, Gaudin, and Mehta.