Statistical Theory of the Energy Levels of Complex Systems. III

A systematic method is developed for calculating the n‐level correlation‐function R_n (x₁, ⋯,x_n), defined as the probability for finding n levels at positions (x₁, ⋯,x_n), regardless of the positions of other levels. It is supposed that the levels of a complex system are statistically equivalent to the eigenvalues of a random symmetric unitary matrix of order N≫n, according to the general theory described in an earlier paper. The 2‐level correlation‐function is found to be

$$\begin{array}{c}{R}_2{\mathrm{(x}}_1{\mathrm{,x}}_2{\text{)=1−{s(r)}}}^2-\left\{{\displaystyle {\int }_r^{\infty }}\mathrm{s(t)dt}\right\}\mathrm{\{ds(r)/dr\},}\\ \mathrm{s(r)=[}\sin {\mathrm{(\pi r)/\pi r],\hspace{0.17em}r=|x}}_1{\mathrm{-x}}_2\mathrm{|,}\end{array}$$

the scale of energy being chosen so that the mean level‐spacing is unity. It is shown how this result could in principle be used in order to determine the proportions of levels in two uncorrelated and superimposed series. An analytic expression for the distribution of nearest‐neighbor level‐spacings, discovered by Gaudin and Mehta, is rederived, and a similar expression is found for the distribution of spacings between next‐nearest neighbors. An unexplained identity relates the nearest and next‐nearest neighbor spacing distributions of a system invariant under time‐reversal to the level‐spacing distribution of a system without time‐reversal invariance.