The energy level statistics of Hamiltonian systems between integrability and chaos: The semiclassical limit

During the past decade or so there has been growing theoretical, numerical and experimental support for the Bohigas-Giannoni-Schmit conjecture (1984) on the applicability of the random matrix theories statistics (GOE, GUE) in the classically ergodic quantal Hamiltonian systems. In the classically integrable systems the spectral fluctuations of the corresponding quantal Hamiltonians are well described by the Poissonian statistics. In the present paper we discuss the statistical properties of energy spectra of generic Hamiltonians in the transition region between integrability and ergodicity (KAM systems). We present statistically highly significant evidence for the fractional power law level repulsion (in the non-semiclassical limit, or near semiclassical limit), which is quite well fitted by the Brody distribution. This is now understood and explained theoretically. However, at sufficiently large level spacings, say S > 1, the Berry-Robnik formulae (1984) for the level spacing distribution are found to be adequate. Moreover, higher in the (far-) semiclassical limit (which we are not yet able to reach in billiards but only in the quantized compactified standard map) we are able to demonstrate clearly that the Berry-Robnik (1984) level statistics as a semiclassical theory is the asymptotically exact theory.