Quantum description of a pseudointegrable system: The π/3-rhombus billiard

Publisher Website
Google Scholar
Abstract
The π/3-rhombus billiard is an example of the simplest pseudointegrable system having an invariant integral surface of genus g=2. We examine the fluctuation properties of eigenvalue sequences belonging to the ‘‘pure rhombus’’ modes (the eigenfunctions take nonzero values on the shorter diagonal). The nearest-neighbor spacing statistics follow the Berry-Robnik distribution with a chaotic fraction ν¯ (corresponding to the Liouville measure of the chaotic subspace) equal to 0.8. The spectral rigidity closely agrees with such a partitioning of phase space. The nodal patterns and the path correlation function exhibit irregularity for most of the corresponding eigenfunctions. Though the amplitude distributions for these closely approximate a Gaussian distribution, the spatial correlations do not agree well with the well-known Bessel oscillations. A few eigenfunctions, however, show regularity. These are localized in those regions of configuration space where the bouncing-ball modes form rectangular bands. The Born-Oppenheimer approximation offers a suitable explanation in terms of a confining potential, and the agreement between the exact and adiabatic eigenvalues improve at higher energies. On the basis of these observations, it turns out that the quantities ν and ν¯ are, in fact, the fractions of regular and irregular states in the eigenvalue sequence under consideration. Thus irregular eigenfunctions do occur for systems with zero Kolmogorov entropy, and the eigenvalue sequence corresponding to these yield Gaussian orthogonal ensemble statistics.