Quantization of generic chaotic 3D billiard with smooth boundary I: energy level statistic

Abstract

Numerical calculation and analysis of extremely high-lying energy spectra, containing thousands of levels with sequential quantum number up to 62,000 per symmetry class, of a generic chaotic 3D quantum billiard is reported. The shape of the billiard is given by a simple and smooth de formation of a unit sphere which gives rise to (almost) fully chaotic classical dynamics. We present an analysis of (i) quantum length spectrum whose smooth part agrees with the 3D Weyl formula and whose oscillatory part is peaked around the periods of classical periodic orbits, (ii) nearest neighbor level spacing distribution and (iii) number variance. Although the chaotic classical dynamics quickly and uniformly explores almost entire energy shell, while the measure of the regular part of phase space is insignificantly small, we find small but significant deviations from GOE statistics which are explained in terms of localization of eigenfunctions onto lower dimensional classically invariant manifolds.