We study the quantum mechanics of the kicked rotator in the classically fully chaotic regime, in the domain of the semiclassical behaviour. We use Izrailev's N-dimensional model for various N ( 4000, which in the limit of large N tends to the quantized kicked rotator, not only for K=5 as studied previously by Izrailev, but for many different values of the classical kick parameter 5( K (35, and also for many different values of the quantum parameter k within [5,60]. We describe the features of dynamical localization as a paradigm for other both time-periodic and time-independent (autonomous) fully chaotic or/and mixed type (generic) Hamiltonian systems. We also generalize the scaling variable lambda =L/N to the case of anomalous diffusion in the classical phase space, by deriving the semiclassical formula for the localization length L, and study the generalized classical diffusion in the system, especially in the regime where the simple minded theory of the normal diffusion fails. We greatly improve the accuracy of the numerical calculations, giving rise to the following conclusions: (C1) The level spacing distribution of the eigenphases (or quasienergies) is very well described by the Brody distribution, systematically better than by other proposed models, for various Brody exponents beta from 0 (Poisson) to 1 (GOE/COE), depending on the strength of the dynamical localization. In the limit of large N and fixed finite localization length L we have always Poisson, even if we are in the classically fully chaotic regime. (C2) We study the eigenfunctions of the Floquet operator and characterize their localization properties using the information entropy measure describing the degree of dynamical localization of the eigenfunctions, which after normalization is given by B on the interval [0,1]. The level repulsion parameter beta and B are almost linearly related, close to the identity line. (C3) We show the existence of a scaling law between B and the relative localization length lambda, now including the regimes of anomalous diffusion. (C4) The above findings are important in a broader perspective, because we have evidence that a similar analysis of the dynamical localization applies also in time-independent (autonomous) Hamilton systems, like in mixed type billiards (Batistić and Robnik 2010,2012), where the Brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates including in the most pronounced regime with beta around 0.5, well away from Poisson and GOE/COE.
We study the quantum mechanics of a billiard (Robnik 1983) in the regime of mixed-type classical phase space (the shape parameter lambda=0.15) at very high-lying eigenstates, starting at about 1.000.000th eigenstate and including the consecutive 587654 eigenstates. By calculating the normalized Poincare Husimi functions of the eigenstates and comparing them with the classical phase space structure, we introduce the overlap criterion which enables us to separate with great accuracy and reliability the regular and chaotic eigenstates, and the corresponding energies. The chaotic eigenstates appear all to be dynamically localized, meaning that they do not occupy uniformly the entire available chaotic classical phase space component, but are localized on a proper subset. We find with unprecedented precision and statistical significance that the level spacing distribution of the regular levels obeys the Poisson statistics, and the chaotic ones obey the Brody statistics, as anticipated in a recent paper by Batistić and Robnik (2010), where the entire spectrum was found to obey the BRB statistics. There are no effects of dynamical tunneling in this regime, due to the high energies, as they decay exponentially with the inverse effective Planck constant which is proportional to the square root of the energy.
We study dynamically localized chaotic eigenstates in the finite dimensional quantum kicked rotator as a paradigm of Floquet systems and in a billiard system of the mixed-type (Robnik 1983) as a paradigm of time-independent Hamilton systems. In the first case we study the spectrum of quasienergies, in the second one the energy spectrum. In the kicked rotator we work in the entirely chaotic regime at K=7, whilst in the billiard we use the Poincare Husimi functions (on the Poincare Birkhoff surface of section) to separate the regular and chaotic eigenstates, and then perform the analysis of 587654 high-lying chaotic eigenstates (starting at about 1.000.000 above the ground state). In both cases we show that the Brody distribution excellently describes the level spacing distribution, with an unprecedented accuracy and statistical significance. The Berry-Robnik picture of separating the regular and chaotic levels in the case of the billiard is also fully confirmed.