The Classical Limit of quantum Evolution in Chaotic Systems

The singular behavior of quantum evolution in the classical limit is of particular interest for chaotic systems. The limit of long time behavior was studied for such systems. It is of special interest in the view of recent calculations of energy correlations, where deviations from Random Matrix Theory (RMT) related to the long time behavior of the classical systems were found. Specifically the semiclassical limit of the evolution operator of the Wigner function for bound systems was investigated. For this purpose coarse graining was introduced. It was argued in general, and tested numerically for the baker map, that in the limit of vanishing coarse graining the spectrum of the evolution operator consists of the Ruelle resonances, that are related to relaxation rates of disturbances of the classical invariant density