Semiclassical energy level statistics in the transition region between integrability and chaos: transition from Brody-like to Berry-Robnik behaviour

We study the energy level statistics of the generic Hamiltonian systems in the transition region between integrability and chaos and present the theoretical and numerical evidence that in the ultimate (far) semiclassical limit the Berry-Robnik (1984) approach is the asymptotically exact theory. However, before reaching that limit, one observes phenomenologically a quasi-universal behaviour characterized by the fractional power-law level repulsion and globally quite well described by the Brody (or Izrailev) distribution. We offer theoretical arguments explaining this extremely slow transition and demonstrate it numerically in improved statistics of the Robnik billiard and in the standard (Chirikov) map on a torus.