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2-point Spectral Correlations for the Square Billiard

Keating, J.P.; Connors, R.D.

December 4, 1996

Keyword(s): spectral correlations; quantum billiards; Poisson statistics

Abstract: We investigate the 2-point correlations in the quantum spectrum of the square billiard. This system is unusual in that the degeneracy of the energy levels increases in the semiclassical limit in such a way that the average level separation is not given by the inverse of the mean density of states. Hence, for example, the standard level spacings distribution does not tend to the Poissonian limit expected for integrable systems. We here calculate the leading- order asymptotic form of a degeneracy-weighted 2-point correlation function using a combination 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 and the second representing a small correction which decays as the inverse of the correlation distance. This is confirmed by numerical computations.

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