2-POINT SPECTRAL CORRELATIONS FOR THE SQUARE BILLIARD

We investigate the two-point correlations in the quantum spectrum of t he square billiard. This system is unusual in that the degeneracy of t he energy levels increases in the semiclassical limit in such a way th at the average level separation is not given by the inverse of the mea n density of states. Hence, for example, the standard level spacings d istribution does not tend to the Poissonian limit expected for integra ble systems. In this paper we calculate the leading-order asymptotic f orm of a degeneracy-weighted two-point correlation function using a co mbination of probabilistic techniques and classical number theory. The result exhibits number-theoretical fluctuations about a mean which is a sum of two terms: one having the usual (constant) Poissonian form a nd the second representing a small correction which decays as the inve rse of the correlation distance. This is confirmed by numerical comput ations.