Numerical demonstration of the Berry-Robnik level spacing distribution

We offer a clear numerical demonstration of the Berry-Robnik level spacing distribution in a dynamical regime in which regular and irregular regions coexist in classical phase space. In order to achieve this we had to go very deeply into the semiclassical limit, which, so far, has only been possible for an abstract dynamical system, namely the quantized standard map on a torus. We have performed an extensive numerical analysis of the quasi-energy spectra and of the eigenstates in the two-dimensional phase-space representation. We have confirmed the validity of Percival's conjecture(1973) that the eigenstates can be clearly classified either as regular or irregular where the small set of mixed-type states vanishes extremely slowly as we approach the far semiclassical limit. It has been verified that the assumptions (statistical independence) implicit in the Berry-Robnik theory are indeed satisfied giving rise to the observed excellent agreement between theory and experiment. The same high quality agreement is also observed in our comparison of the semiclassical theoretical (Seligman and Verbaarschot, 1985) and numerical delta statistics.