Abstract :
There are three universality classes of spectral fluctuations of the quantal Hamiltonian systems (with a small number of freedoms, say two or 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 present recent numerical results in support of this conjecture, and also discuss 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 body of the paper is devoted to the energy level statistics of the generic systems in the transition region between integrability and chaos (KAM-systems with mixed dynamics in the classical phase-space). We discuss the applicability of the semiclassical theory (Berry-Robnik formulae for the level spacing distribution), and show that it applies at large 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 reliable and statistically significant evidence for the existence of the quasi-universal fractional power law level repulsion, implying surprisingly good analytical fit by the Brody (and Izrailev) distribution. We discuss possible theoretical approaches such as sparse banded random matrix ensembles (SBRME), the Dyson Pechukas-Yukawa picture, and new ideas related to the localization of stationary states (in terms of Wigner functions within the classically invariant ergodic components).
