RECENT DEVELOPMENTS IN ENERGY-LEVEL STATISTICS IN GENERIC SYSTEMS BETWEEN INTEGRABILITY AND CHAOS

During the past decade or so there has been growing theoretical, numer ical and experimental support for the Bohigas-Giannoni-Schmit Conjectu re (1984) on the applicability of the random matrix theories statistic s (GOE, GUE) in the classically ergodic quantal Hamiltonian systems. I n the classically integrable systems the spectral fluctuations of the corresponding quantal Hamiltonians are well described by the Poissonia n statistics. In the present paper we discuss the statistical properti es of energy spectra of generic Hamiltonians in the transition region between integrability and ergodicity (KAM-systems). We present convinc ing statistically highly significant evidence for the fractional power law level repulsion (in the non-semiclassical limit, or near semiclas sical limit), which is quite well fitted by the Brody distribution and even more so by the Izrailev distribution. However, at sufficiently l arge level spacings, say S>1, the Berry-Robnik formulae for the level spacing distribution are found to be adequate. We discuss the possible theoretical approaches and explanations. The phenomenon of power law level repulsion is partially understood in terms of the sparsed banded random matrix ensembles (SBRME).