Topics in quantum chaos of generic systems

TL;DR

This paper reviews quantum chaos in systems with mixed regular and chaotic classical dynamics, discussing spectral fluctuation universality, invariant measure calculation challenges, and the Berry-Robnik model's applicability and crossover phenomena.

Contribution

It provides a comprehensive review of spectral fluctuation classes, invariant measure issues, and the Berry-Robnik picture, including new insights into crossover regimes at finite .

Findings

Spectral fluctuations follow GOE/GUE or Poisson statistics depending on classical dynamics.

Invariant measure calculation remains challenging in mixed systems.

Crossover behavior observed at finite  due to quantum state localization.

Abstract

We review the main ideas and results in the stationary problems of quantum chaos in generic (mixed) systems, whose classical dynamics has regular (invariant tori) and chaotic regions coexisting in the phase space. First we discuss the universality classes of spectral fluctuations (GOE/GUE for ergodic systems, and Poissonian for integrable systems). We explain the problems in the calculation of the invariant (Liouville) measure of classically chaotic components, which has recently been studied by Robnik et al (1997) and by Prosen and Robnik (1998). Then we describe the Berry-Robnik (1984) picture, which is claimed to become exact in the strict semiclassical limit $\hbar\to 0$. However, at not sufficiently small values of $\hbar$ we see a crossover regime due to the localization properties of stationary quantum states where Brody-like behaviour with the fractional power law level repulsion is observed in the corresponding quantal energy spectra.