Level statistics of a noncompact integrable billiard

A noncompact, nonintegrable billiard of importance in cosmology—the infinite equilateral triangle on a space of constant negative curvature—gives rise to a natural noncompact integrable approximation, which is studied here quantum mechanically. The Weyl formula for the averaged spectral staircase contains a nonstandard logarithmic term due to the noncompactness of the billiard. For two symmetry classes we determine numerically the spectral staircase, the level-spacing distribution p(s), and the averaged ${\mathrm{\Delta }}_3$ statistic, and compare with analytical expressions that follow from Poisson statistics and semiclassical theory. The Poissonian result for the correlation function of ${\mathrm{\Delta }}_3$ is derived in closed form and also compared with the data.