We perform an extensive and detailed analysis of the generalized diffusion processes in deterministic area preserving maps with noncompact phase space, exemplified by the standard map, with the special emphasis on understanding the anomalous diffusion arising due to the accelerator modes. The accelerator modes and their immediate neighborhood undergo ballistic transport in phase space, and also the greater vicinity of them is still much affected (``dragged'') by them, giving rise to the non-Gaussian (accelerated) diffusion. The systematic approach rests upon the following applications: the GALI method to detect the regular and chaotic regions and thus to describe in detail the structure of the phase space, the description of the momentum distribution in terms of the L\'evy stable distributions, and the numerical calculation of the diffusion exponent and of the corresponding diffusion constant. We use this approach to analyze in detail and systematically the standard map at all values of the kick parameter $K$, up to $K=70$. All complex features of the anomalous diffusion are well understood in terms of the role of the accelerator modes, mainly of period 1 at large $K$\ge${}2$\pi${}$, but also of higher periods ($2,3,4,...$) at smaller values of $K$\le${}2$\pi${}$.

The quantum localization is one of the remarkable phenomena in the studies of quantum chaos and plays an important role in various contexts. Thus, an understanding of the properties of quantum localization is essential. In spite of much effort dedicated to investigating the manifestations of localization in the time-dependent systems, the features of localization in time-independent systems are still less explored, particularly in quantum systems which correspond to the classical systems with smooth Hamiltonian. In this work, we present such a study for a quantum many-body system, namely, the Dicke model. The classical counterpart of the Dicke model is given by a smooth Hamiltonian with two degrees of freedom. We examine the signatures of localization in its chaotic eigenstates. We show that the entropy localization measure, which is defined in terms of the information entropy of Husimi distribution, behaves linearly with the participation number, a measure of the degree of localization of a quantum state. We further demonstrate that the localization measure probability distribution is well described by the $\beta$ distribution. We also find that the averaged localization measure is linearly related to the level repulsion exponent, a widely used quantity to characterize the localization in chaotic eigenstates. Our findings extend the previous results in billiards to the quantum many-body system with classical counterpart described by a smooth Hamiltonian, and they indicate that the properties of localized chaotic eigenstates are universal.

We study the kicked rotator in the classically fully chaotic regime using Izrailev's $N$-dimensional model for various $N\le 4000$, which in the limit $N\to \infty$ tends to the quantized kicked rotator. We do treat not only the case $K=5$, as studied previously, but also many different values of the classical kick parameter $5\le K\le 35$ and many different values of the quantum parameter $k\in [5,60]$. We describe the features of dynamical localization of chaotic eigenstates as a paradigm for other both time-periodic and time-independent (autonomous) fully chaotic or/and mixed-type Hamilton systems. We generalize the scaling variable $\Lambda =l_{\infty }/N$ to the case of anomalous diffusion in the classical phase space by deriving the localization length $l_{\infty }$ for the case of generalized classical diffusion. We greatly improve the accuracy and statistical significance of the numerical calculations, giving rise to the following conclusions: (1) 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 ${\beta }_{\mathrm{BR}}$. (2) We study the eigenfunctions of the Floquet operator and characterize their localization properties using the information entropy measure, which after normalization is given by ${\beta }_{\mathrm{loc}}$ in the interval $[0,1]$. The level repulsion parameters ${\beta }_{\mathrm{BR}}$ and ${\beta }_{\mathrm{loc}}$ are almost linearly related, close to the identity line. (3) We show the existence of a scaling law between ${\beta }_{\mathrm{loc}}$ and the relative localization length $\Lambda$, now including the regimes of anomalous diffusion. The above findings are important also for chaotic eigenstates in time-independent systems [Batistić and Robnik, J. Phys. A: Math. Gen. 43, 215101 (2010); arXiv:1302.7174 (2013)], where the Brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates, even in the mixed-type systems (after separation of regular and chaotic eigenstates).

We construct an explicit model of the gravity trapping domain-wall potential, where for the first time we can study explicitly the graviton wave function fluctuations for any thickness of domain wall. A concrete form of the potential depends on one parameter $0\le x\le \frac{\pi }{2}$, which effectively parameterizes the thickness of the domain wall with specific limits $x\to 0$ and $x\to \frac{\pi }{2}$ corresponding to the thin and the thick wall, respectively. The analysis of continuum Kaluza-Klein fluctuations yields explicit expressions for both small and large Kaluza-Klein energy. We also derive specific explicit conditions in the regime $x>1$, for which the fluctuation modes exhibit a resonance behavior, and which could sizably affect the modifications of the four-dimensional Newton’s law at distances that typically are by 4 orders of magnitude larger than those relevant for Newton’s law modifications of thin walls.

Some dynamical properties for a dissipative kicked rotator are studied. Our results show that when dissipation is taken into account a drastic change happens in the structure of the phase space in the sense that the mixed structure is modified and attracting fixed points and chaotic attractors are observed. A detailed numerical investigation in a two-dimensional parameter space based on the behavior of the Lyapunov exponent is considered. Our results show the existence of infinite self-similar shrimp-shaped structures corresponding to periodic attractors, embedded in a large region corresponding to the chaotic regime.