Systematic Characterization of mth-Order Energy-Level Spacing Distributions

Abstract

The mth‐order energy‐level spacing distributions p^(m)(x) for complex spectra are defined in terms of a general joint probability distribution P_N(λ₁, … λ_N) for N consecutive eigenvalues. The precise limiting processes involved are explained, and are subsequently used to obtain two formal representations of p^(m)(x). Both representations yield p^(m)(x) = x^m/m! exp (−x) for statistically independent eigenvalues. One of the representations, which is an extension of Dyson's method for m = 0, 1, is applied to the superposition of n independent sequences of levels. General asymptotic results are found for the mth‐order distributions for (a) small x, arbitrary n, and (b) arbitrary x with n → ∞.