Numerical verification of Percival's conjecture in a quantum billiard

In order to verify Percival's conjecture [J. Phys. B 6,L229 (1973)] we study a planar billiard in its classical and quantum versions. We provide an evaluation of the nearest-neighbor level-spacing distribution for the Cassini oval billiard, taking into account relations with classical results. The statistical behavior of integrable and ergodic systems has been extensively confirmed numerically, but that is not the case for the transition between these two extremes. Our system's classical dynamics undergoes a transition from integrability to chaos by varying a shape parameter. This feature allows us to investigate the spectral fluctuations, comparing numerical results with semiclassical predictions founded on Percival's conjecture. We obtain good $global$ agreement with those predictions, in clear contrast with similar comparisons for other systems found in the literature. The structure of some eigenfunctions, displayed in the quantum Poincar\'e section, provides a clear explanation of the conjecture.