Abstract
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.
Highlights
In 1973 Percival conjectured that in the semiclassical limit, the spectrum of a generic dynamical system consists of two parts with strongly contrasting properties: a regular and an irregular part [1]
To verify Percival’s conjecture, we have studied the quantum version of a billiard, depending on one shape parameter d
The level spacing statistics was fitted by two distributions depending on one parameter: the semiclassical Berry-Robnik distribution (BRD), which is founded on Percival’s conjecture, and the Brody distribution (BD)
Summary
In 1973 Percival conjectured that in the semiclassical limit, the spectrum of a generic dynamical system consists of two parts with strongly contrasting properties: a regular and an irregular part [1]. The other special case corresponds to mixing systems where almost all orbits explore densely and chaoticaly the energy surface In this case, Bohigas et al [3] conjectured that the fluctuation properties of these spectra can be modeled by the ensemble of random real symmetric matrices [the Gaussian Orthogonal Ensemble (GOE)] [4]. Berry and Robnik [6], based on Percival’s conjecture, considered independent sequences of levels associated with each connected regular or irregular classical phase-space region. In 1994 Prosen and Robnik [19] confirmed numerically semiclassical predictions (the BRD) working on an abstract dynamical system : the standard map on a torus To agree with this theory, they needed to compute extremely high excited states (around the 30 106).
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