SEMICLASSICAL ENERGY-LEVEL STATISTICS THE TRANSITION REGION BETWEEN INTEGRABILITY AND CHAOS - TRANSITION FROM BRODY-LIKE TO BERRY-ROBNIK BEHAVIOR

We study the energy level statistics of the generic Hamiltonian system s in the transition region between integrability and chaos and present the theoretical and numerical evidence that in the ultimate (far) sem iclassical limit the Berry-Robnik (1984) approach is the asymptoticall y exact theory. However, before reaching that limit, one observes phen omenologically a quasi-universal behaviour characterized by the fracti onal power-law level repulsion and globally quite well described by th e Brody (or Izrailev) distribution. We offer theoretical arguments exp laining this extremely slow transition and demonstrate it numerically in improved statistics of the Robnik billiard and in the standard (Chi rikov) map on a torus.