Abstract

The known manifestation of quantum chaos is the so-called Wigner-Dyson distribution P(s) for spacings between neighbouring levels in the spectrum. In the other limiting case of completely integrable systems, the distribution P(s) turns out to be very close to the Poissonian one. In the present chapter, the influence of quantum effects on level statistics is studied for the case in which the corresponding classical systems are fully chaotic. The numerical study of the kicked rotator model with a finite number of states allows us to establish the link between the degree of localization and level repulsion. As a good model for this phenomenon of localization, the band random matrices are suggested. It is shown that such matrices can be used to describe statistical properties of localized quantum chaos.

Introduction

One of the important problems in the theory of quantum chaos is the description of statistical properties of systems using classical and quantum (semiclassical) parameters. Numerous studies (see the review [1] and references therein) have shown that the general situation is very complicated and no universal properties can be predicted when in the classical limit the motion is not fully chaotic. On the other hand, it has been proved numerically that for many classical models (for example, billiards, see [2]) with strong chaos, statistical properties both of energy (quasienergy) spectra and eigenfunctions are well described by the Random Matrix Theory (RMT) [3].