Quantum Chaos in Generic Systems

First I briefly review the basic elements of the stationary quantum chaos in Hamiltonian systems, the universality classes of energy spectra and eigenfunctions. Then I consider the problem of the generic systems whose classical dynamics and the phase portrait is of the mixed type, i.e. regular for certain initial conditions and irregular (chaotic) for other initial conditions. I present the Berry-Robnik picture, the Principle of Uniform Semiclassical Condensation (of the Wigner functions of the eigenstates), and the statistical description of the energy spectra in terms of E(k,L) statistics, which is shown to be valid in the semiclassical limit of sufficiently small effective Planck constant and is numerically firmly confirmed. Then I consider the spectral autocorrelation function and the form factor (its Fourier transform) in the same limit, and show its agreement with the numerical investigations in the regular and fully chaotic cases. I show the numerical evidence for the deviations from that prediction in mixed type systems at low energies, due to localization and tunneling effects. Here are also the important open theoretical questions that I address.