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We study the quantum kicked rotator in the classically fully chaotic regime $K=10$ and for various values of the quantum parameter $k$ using Izrailev's $N$-dimensional model for various $N \le 3000$, which in the limit $N \rightarrow \infty$ tends to the exact quantized kicked rotator. By numerically calculating the eigenfunctions in the basis of the angular momentum we find that the localization length ${\cal L}$ for fixed parameter values has a certain distribution, in fact its inverse is Gaussian distributed, in analogy and in connection with the distribution of finite time Lyapunov exponents of Hamilton systems. However, unlike the case of the finite time Lyapunov exponents, this distribution is found to be independent of $N$, and thus survives the limit $N=\infty$. This is different from the tight-binding model of Anderson localization. The reason is that the finite bandwidth approximation of the underlying Hamilton dynamical system in the Shepelyansky picture (D.L. Shepelyansky, {\em Phys. Rev. Lett.} {\bf 56}, 677 (1986)) does not apply rigorously. This observation explains the strong fluctuations in the scaling laws of the kicked rotator, such as e.g. the entropy localization measure as a function of the scaling parameter $\Lambda={\cal L}/N$, where $\cal L$ is the theoretical value of the localization length in the semiclassical approximation. These results call for a more refined theory of the localization length in the quantum kicked rotator and in similar Floquet systems, where we must predict not only the mean value of the inverse of the localization length $\cal L$ but also its (Gaussian) distribution, in particular the variance. In order to complete our studies we numerically analyze the related behavior of finite time Lyapunov exponents in the standard map and of the 2$\times$2 transfer matrix formalism. This paper is extending our recent work., Comment: 12 pages, 9 figures (accepted for publication in Physical Review E). arXiv admin note: text overlap with arXiv:1301.4187

We highlight a few recent results on the effect of the diffusion process in deterministic area preserving maps with noncompact phase space, namely the standard map. In more detail, we focus on the anomalous diffusion arising due to the accelerator modes, i.e. resonant-like features of the phase space which are transported in a super-diffusive (ballistic) manner. Their presence affects also the trajectories lying in the immediate neighborhood resulting in anomalous (non-Gaussian) diffusion. We aim to shed some light on these special properties of the phase space by utilizing the diffusion exponent (rate of diffusion) and the momentum distribution in terms of the L\'evy stable distributions, with the goal to reach an understanding of the global behaviour. To this end we consider a rather large interval for the kick parameter ($0\le K \le 70$) of the standard map where accelerator modes of different periodicity exist., Comment: 5 pages, 3 figures, proceedings of the Conference "New Challenges in Complex Systems Science", Waseda University, Tokyo, Japan, October 2014

We study the fundamental question of dynamical tunneling in generic two-dimensional Hamiltonian systems by considering regular-to-chaotic tunneling rates. Experimentally, we use microwave spectra to investigate a mushroom billiard with adjustable foot height. Numerically, we obtain tunneling rates from high precision eigenvalues using the improved method of particular solutions. Analytically, a prediction is given by extending an approach using a fictitious integrable system to billiards. In contrast to previous approaches for billiards, we find agreement with experimental and numerical data without any free parameter., Comment: 4 pages, 4 figures

We study the level spacing distribution $P(S)$ of 2D real random matrices both symmetric as well as general, non-symmetric. In the general case we restrict ourselves to Gaussian distributed matrix elements, but different widths of the various matrix elements are admitted. The following results are obtained: An explicit exact formula for $P(S)$ is derived and its behaviour close to S=0 is studied analytically, showing that there is linear level repulsion, unless there are additional constraints for the probability distribution of the matrix elements. The constraint of having only positive or only negative but otherwise arbitrary non-diagonal elements leads to quadratic level repulsion with logarithmic corrections. These findings detail and extend our previous results already published in a preceding paper. For the {\em symmetric} real 2D matrices also other, non-Gaussian statistical distributions are considered. In this case we show for arbitrary statistical distribution of the diagonal and non-diagonal elements that the level repulsion exponent $\rho$ is always $\rho = 1$, provided the distribution function of the matrix elements is regular at zero value. If the distribution function of the matrix elements is a singular (but still integrable) power law near zero value of $S$, the level spacing distribution $P(S)$ is a fractional exponent pawer law at small $S$. The tail of $P(S)$ depends on further details of the matrix element statistics. We explicitly work out four cases: the constant (box) distribution, the Cauchy-Lorentz distribution, the exponential distribution and, as an example for a singular distribution, the power law distribution for $P(S)$ near zero value times an exponential tail., Comment: 21 pages, no figures, submitted to Zeitschrift fuer Naturforschung A

The power law $1/f^{\alpha}$ in the power spectrum characterizes the fluctuating observables of many complex natural systems. Considering the energy levels of a quantum system as a discrete time series where the energy plays the role of time, the level fluctuations can be characterized by the power spectrum. Using a family of quantum billiards, we analyze the order to chaos transition in terms of this power spectrum. A power law $1/f^{\alpha}$ is found at all the transition stages, and it is shown that the exponent $\alpha$ is related to the chaotic component of the classical phase space of the quantum system., Comment: 4 pages, 5 figures, accepted for publication in Phys. Rev. Lett