Beyond the Berry–Robnik regime: a random matrix study of tunneling effects

Abstract

We consider the energy spectra of quantum Hamilton systems whose classical dynamics is of mixed type, i.e. regular for some initial conditions in the classical phase space and chaotic for the complementary initial conditions. In the strict semiclassical limit when the effective Planck constant ℏeff is sufficiently small the Berry–Robnik (BR) statistics applies, while at larger values of ℏeff (or smaller energies) one sees deviations from BR due to localization and tunneling effects. We derive a two-level random matrix model, which describes these effects and can be treated analytically in a closed form. The coupling between the regular and chaotic levels due to tunneling is assumed to be Gaussian distributed. This two-level model describes most of the features of matrices of large dimensions (here N = 1000), which we treat numerically, and is predicted to apply in mixed type systems at low energies. The proposed analytical level spacing distribution function has two parameters, the BR parameter ρ, characterizing the classical phase space, and the coupling (antenna distortion or tunneling) parameter σ between states. Localization effects so far are not included in our analysis except in the special case where ρ can be replaced by some effective ρeff.