Quantum Spectra and Transition from Regular to Chaotic Classical Motion

We compute numerically the eigenvalues of a family of two-dimensional Hamiltonians which give rise to regular or chaotic classical motion depending on the particular choice of parameters. A close relationship is found between the spectral statistics and the fraction of classical phase space covered by chaotic trajectories. In the extreme regular and chaotic cases the system displays Poisson and Gaussian-orthogonal-ensemble statistics, respectively. A one-parameter random-matrix model is proposed to describe the spectral statistics in intermediate situations.