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We look at the high-lying eigenstates (from the 10,001st to the 13,000th) inthe Robnik billiard (defined as a quadratic conformal map of the unit disk)with the shape parameter $\lambda=0.15$. All the 3,000 eigenstates have beennumerically calculated and examined in the configuration space and in the phasespace which - in comparison with the classical phase space - enabled a clearcut classification of energy levels into regular and irregular. This is thefirst successful separation of energy levels based on purely dynamical ratherthan special geometrical symmetry properties. We calculate the fractionalmeasure of regular levels as $\rho_1=0.365\pm 0.01$ which is in remarkableagreement with the classical estimate $\rho_1=0.360\pm 0.001$. This findingconfirms the Percival's (1973) classification scheme, the assumption inBerry-Robnik (1984) theory and the rigorous result by Lazutkin (1981,1991). Theregular levels obey the Poissonian statistics quite well whereas the irregularsequence exhibits the fractional power law level repulsion and globallyBrody-like statistics with $\beta = 0.286\pm0.001$. This is due to the stronglocalization of irregular eigenstates in the classically chaotic regions.Therefore in the entire spectrum we see that the Berry-Robnik regime is not yetfully established so that the level spacing distribution is correctly capturedby the Berry-Robnik-Brody distribution (Prosen and Robnik 1994).Comment: 20 pages, file in plain LaTeX, 7 figures upon request submitted to J.  Phys. A. Math. Gen. in December 199

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