PERIODIC-ORBIT THEORY OF THE NUMBER VARIANCE SIGMA(2)(L) OF STRONGLY CHAOTIC SYSTEMS

We discuss the number variance Sigma(2)(L) and the spectral form factor F(tau) of the energy levels of bound quantum systems whose classical counterparts are strongly chaotic. Exact periodic-orbit representations of Sigma(2) (L) and F(tau) are derived which explain the breakdown of universality, i.e., the deviations from the predictions of random-matrix theory. The relation of the exact spectral form factor F(tau) to the commonly used approximation K(tau) is clarified. As an illustration the periodic-orbit representations are tested in the case of a strongly chaotic system at low and high energies including very long-range correlations up to L = 700. Good agreement between ''experimental'' data and theory is obtained.