RECENT DEVELOPMENTS IN THE ENERGY-LEVEL STATISTICS IN GENERIC SYSTEMSBETWEEN INTEGRABILITY AND CHAOS

There are three universality classes of spectral fluctuations of the q uantal Hamiltonian systems (with a small number of freedoms, say two o r more). In classically integrable quantal systems Poisson statistics apply, whilst in classically ergodic systems the random matrix theory applies: GOE if there is an antiunitary symmetry, and GUE if there is no antiunitary symmetry (such as the time reversal symmetry). We prese nt recent numerical results in support of this conjecture, and also di scuss the theoretical arguments. In particular, we raise the question of the adequacy and accuracy of the semiclassical approximations such as the torus quantization (EBK) and the Gutzwiller theory. The main bo dy of the paper is devoted to the energy level statistics of the gener ic systems in the transition region between integrability and chaos (K AM-systems with mixed dynamics in the classical phase-space). We discu ss the applicability of the semiclassical theory (Berry-Robnik formula e for the level spacing distribution), and show that it applies at lar ge spacings in the near semiclassical limit, and is certainly expected to apply for all spacings in the far (= strict) semiclassical limit. In the near semiclassical limit and at small spacings we present relia ble and statistically significant evidence for the existence of the qu asi-universal fractional power law level repulsion, implying surprisin gly good analytical fit by the Brody (and Izrailev) distribution. We d iscuss possible theoretical approaches such as sparse banded random ma trix ensembles (SBRME), the Dyson-Pechukas-Yukawa picture, and new ide as related to the localization of stationary states (in terms of Wigne r functions within the classically invariant ergodic components).