Periodic-orbit theory of the number variance Σ2( L) of strongly chaotic systems

Abstract

We discuss the number variance Σ 2( L) and the spectral form factor F( τ) of the energy levels of bound quantum systems whose classical counterparts are strongly chaotic. Exact periodic-orbit representations of Σ 2( L) and F( τ) 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( τ) to the commonly used approximation K( τ) 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.