Recent Advances in Quantum Chaos of Generic Systems

Abstract

We review the fundamental concepts of quantum chaos in Hamiltonian systems, as the basis to understand the properties of generic systems (so-called mixed-type systems, or systems with divided phase space), where the classical regular regions and chaotic regions coexist in the phase space. The quantum evolution of classically chaotic bound systems does not possess the sensitive dependence on initial conditions, and thus no chaotic behavior occurs, whereas the study of the stationary solutions of the Schrödinger equation in the quantum phase space (Wigner functions) reveals precise analogy of the structure of the classical phase portrait. We analyze the regular eigenstates associated with invariant tori in the classical phase space, and the chaotic eigenstates associated with the classically chaotic regions, and the corresponding energy spectra. Then we present the theoretical description of the generic (mixed-type) systems for the case of no quantum localization of the chaotic eigenstates (all states being extended) introduced by Berry and Robnik (J Phys A Math Gen 17:2413, 1984). The effects of quantum localization of the chaotic eigenstates set in when the Heisenberg time scale is shorter than the classical transport time (such as diffusion time), and these effects are treated phenomenologically, resulting in Brody-like level statistics for the chaotic states alone, which can be found also at very high-lying levels. Then we treat the level spacing distribution for the combined (total) energy spectrum of a generic system. The coupling between the regular and the irregular eigenstates due to tunneling, and of the corresponding levels, manifests itself only in low-lying levels, because it decreases exponentially with increasing energy (or inverse effective Planck constant).