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- dynamical systems (2)
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A formalism is developed for calculating the quasienergy states and spectrum for time-periodic quantum systems when a time-periodic dynamical invariant operator with a nondegenerate spectrum is known. The method, which circumvents the integration of the Schr-odinger equation, is applied to an integrable class of systems, where the global invariant operator is constructed. Furthermore, a local integrable approximation for more general non-integrable systems is developed. Numerical results are presented for the doubleresonance model.

We present an entropy concept measuring quantum localization in dynamical systems based on time averaged probability densities. The suggested entropy concept is a generalization of a recently introduced [PRL 75, 326 (1995)] phase-space entropy to any representation chosen according to the system and the physical question under consideration. In this paper we inspect the main characteristics of the entropy and the relation to other measures of localization. In particular the classical correspondence is discussed and the statistical properties are evaluated within the framework of random vector theory. In this way we show that the suggested entropy is a suitable method to detect quantum localization phenomena in dynamical systems.

For periodically driven systems, quantum tunneling between classical resonant stability islands in phase space separated by invariant KAM curves or chaotic regions manifests itself by oscillatory motion of wave packets centered on such an island, by multiplet splittings of the quasienergy spectrum, and by phase space localisation of the quasienergy states on symmetry related ,ux tubes. Qualitatively di,erent types of classical resonant island formation | due to discrete symmetries of the system | and their quantum implications are analysed by a (uniform) semiclassical theory. The results are illustrated by a numerical study of a driven non-harmonic oscillator.