T. Prosen et M. Robnik, ENERGY-LEVEL STATISTICS IN THE TRANSITION REGION BETWEEN INTEGRABILITY AND CHAOS, Journal of physics. A, mathematical and general, 26(10), 1993, pp. 2371-2387
A generic one-parameter family of billiards discovered and introduced
by Robnik (1983) is used to study the spectral properties of correspon
ding quantum systems. When the parameter is varied a smooth transition
from an integrable system over a typical KAM system to an almost ergo
dic system can be observed. We calculate up to 7600 lowest reliable en
ergy levels. A detailed analysis of the numerical data shows significa
nt deviation from the semiclassical Berry-Robnik formulae for the near
est-neighbour level spacing distribution P(S) except for large level s
pacings, S > 1, which can only be explained by a very slow convergence
towards the semiclassical regime where these formulae are predicted t
o be correct. At small S the power-law level repulsion is clearly obse
rved and a fit by the phenomenological formula by Izrailev is statisti
cally significant.