Separating the regular and irregular energy levels and their statistics in a Hamiltonian system with mixed classical dynamics

We look at the high-lying eigenstates (from the 10001 to the 13000) in the Robnik billiard (defined as a quadratic conformal map of the unit disk) with the shape parameter lambda =0.15. All the 3000 eigenstates have been numerically calculated and examined in the configuration space and in the phase space which-in comparison with the classical phase space-enabled a clear cut classification of energy levels into regular and irregular. This is the first successful separation of energy levels based on purely dynamical rather than special geometrical symmetry properties. We calculate the fractional measure of regular levels as rho 1 =0.365+or-0.01, which is in remarkable agreement with the classical estimate rho 1 =0.360+or-0.001. This finding confirms the Percival's (1973) classification scheme, the assumption in Berry-Robnik (1984) theory and the rigorous result by Lazutkin (1981, 1991). The regular levels obey the Poissonian statistics quite well, whereas the irregular sequence exhibits the fractional power-law level repulsion and globally Brody-like statistics with beta =0.286+or-0.001. This is due to the strong localization of irregular eigenstates in the classically chaotic regions. Therefore, in the entire spectrum we see that the Berry-Robnik regime is not yet fully established so that the level spacing distribution is correctly captured by the Berry-Robnik-Brody distribution.