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We study the quantum mechanics of a billiard (Robnik 1983) in the regime ofmixed-type classical phase space (the shape parameter \lambda=0.15) at veryhigh-lying eigenstates, starting at about 1.000.000th eigenstate and includingthe consecutive 587654 eigenstates. By calculating the normalized Poincar\'eHusimi functions of the eigenstates and comparing them with the classical phasespace structure, we introduce the overlap criterion which enables us toseparate with great accuracy and reliability the regular and chaoticeigenstates, and the corresponding energies. The chaotic eigenstates appear allto be dynamically localized, meaning that they do not occupy unformly theentire available chaotic classical phase space component, but are localized ona proper subset. We find with unprecedented precision and statisticalsignificance that the level spacing distribution of the regular levels obeysthe Poisson statistics, and the chaotic ones obey the Brody statistics, asanticipated in a recent paper by Batisti\'c and Robnik (2010), where the entirespectrum was found to obey the BRB statistics. There are no effects ofdynamical tunneling in this regime, due to the high energies, as they decayexponentially with the inverse effective Planck constant which is proportionalto the square root of the energy.

Dynamical localization of chaotic eigenstates in the mixed-type systems:  spectral statistics in a billiard system after separation of regular and  chaotic eigenstates