Statistical properties of many-particle spectra

Many-particle eigenvalue densities, generated by k-body interactions in asymptotically large spectroscopic spaces, are studied in terms of the density moments of a Gaussian orthogonal ensemble of such interactions. It is shown that the densities approach Gaussian as particle number increases; the rate of approach is determined, as is the rate of departure from the semicircular form which obtains in the few-particle limit, as well as the low moment representation for intermediate cases. By considering the variances of the moments it is shown that a convergence in probability of the individual densities to the ensemble-averaged density, first demonstrated by Grenander for the few-particle limit, is valid in general. In part as a preliminary to a study of the density and spectrum fluctuations, explicit forms are derived for the covariances of low order moments for arbitrary particle number and of moments of general order in the few-particle and many-particle limits. In the limiting cases the ensemble eigenvalue densities are exhibited in terms of a set of statistically uncorrelated Chebyshev or Hermite-polynomial excitations of the ensemble-averaged densities. Some explicit results for GOE fluctuations are given