Length spectrum and semiclassical density of states for an almost-integrable billiard system.

Abstract

By obtaining analytically the length spectrum of periodic orbits, a semiclassical analysis is carried out for the 1/3\ensuremath{\pi}-rhombus billiard system, which is the simplest possible example of a class of the almost-integrable billiard system. A complete set of length spectra enables us to examine the difference of the semiclassical density of states between the almost-integrable (even-parity states) and completely integrable (odd-parity states) case. Within the framework of the ordinary periodic orbit expansion including only families of periodic orbits, the universality of the quadratic long-range level statistics for the present almost-integrable billiard system cannot be discriminated qualitatively from that for the corresponding integrable billiard system. Nevertheless, an extensive numerical study for the nearest-neighbor level-spacing distribution and the spectral rigidity demonstrates that there exists level repulsion in the almost-integrable case.