Energy level statistics in the transition region between integrability and chaos

A generic one-parameter family of billiards discovered and introduced by Robnik (1983) is used to study the spectral properties of corresponding quantum systems. When the parameter is varied a smooth transition from an integrable system over a typical KAM system to an almost ergodic system can be observed. The authors calculate up to 7600 lowest reliable energy levels. A detailed analysis of the numerical data shows significant deviation from the semiclassical Berry-Robnik formulae for the nearest-neighbour level spacing distribution P(S) except for large level spacings, S>1, which can only be explained by a very slow convergence towards the semiclassical regime where these formulae are predicted to be correct. At small S the power-law level repulsion is clearly observed and a fit by the phenomenological formula by Izrailev (1988,1989) is statistically significant.