Class of Ensembles in the Statistical Theory of Energy-Level Spectra

The investigation of a large class of ensembles in the statistical theory of energy‐level spectra is initiated. Each member of this class is characterized by a joint probability density for N consecutive eigenvalues of the form $$P_{\mathrm{N\beta }}{\mathrm{(\lambda }}_1{\mathrm{,\dots \lambda }}_N{\mathrm{)=\Omega }}_{\mathrm{N\beta }}^{\text{−1}}\left\{{\displaystyle \prod _{\text{i=1}}^N}{\mathrm{\hspace{0.17em}f(\lambda }}_i)\right\}\hspace{0.17em}{\displaystyle \prod _{\mathrm{k<l}}}{\mathrm{\hspace{0.17em}|\lambda }}_k{\mathrm{-\lambda }}_l{|}^{\beta }$$ , where a ≤ λ_i ≤ b, and β may be 1, 2, or 4. Formal calculations of the nearest‐neighbor spacing distribution and the level density are made for β = 2. Results are in terms of asymptotic properties of orthogonal polynomials. It is conjectured that spacing distributions are relatively insensitive to the function f(λ) and the interval [a, b]. When f(λ) = 1 and b = −a = 1, the resulting (Legendre) ensemble has the same spacing distribution as the Gaussian and Dyson ensembles. The level density is concave upward and rapidly increasing for λ ≥ 0, qualitatively resembling actual nuclear and atomic densities. This feature is not present in previously investigated ensembles. Certain invariant matrix ensembles introduced by Dyson, which are of the above type, have the same level density and nearest‐neighbor spacing distribution as the Legendre ensemble.