University of Maribor, Slovenia

Quantum localization of classically chaotic eigenstatets is one of the most important phenomena in quantum chaos, or more generally - wave chaos, along with the characteristic behaviour of statistical properties of the energy spectra. Quantum localization sets in, if the Heisenberg time $t_H$ of the given system is shorter than the classical transport times of the underlying classical system, i.e. when the classical transport is slower than the quantum time resolution of the evolution operator. The Heisenberg time $t_H$, as an important characterization of every quantum system, is namely equal to the ratio of the Planck constant $2\pi\hbar$ and the mean spacing between two nearest energy levels $\Delta E$, $t_H=2\pi \hbar/\Delta E$. We shall show the functional dependence between the degree of localization and the spectral statistics in autonomous (time independent) systems, in analogy with the kicked rotator, which is the paradigm of the time periodic (Floquet) systems, and shall demonstrate the approach and the method in the case of a billiard family in the dynamical regime between the integrability (circle) and full chaos (cardioid), where we shall extract the chaotic eigenstates. The degree of localization is determined by two localization measures, using the Poincar\'e Husimi functions (which are the Gaussian smoothed Wigner functions in the Poincar\'e Birkhoff phase space), which are positive definite and can be treated as quasi-probability densities. The first measure $A$ is defined by means of the information entropy, whilst the second one, $C$, in terms of the correlations in the phase space of the Poincar\'e Husimi functions of the eigenstates. Surprisingly, and very satisfactory, the two measueres are linearly related and thus equivalent. One of the main manifestations of chaos in chaotic eigenstates in absence of the quantum localization is the energy level spacing distribution $P(S)$ (of nearest neighbours), which at small $S$ is linear $P(S)\propto S$, and we speak of the linear level repulsion, while in the integrable systems we have the Poisson statistics (exponential function $P(S)=\exp (-S)$), where there is no level repulsion ($P(0)=1\not= 0$). In fully chaotic regime with quantum localization we observe that $P(S)$ at small $S$ is a power law $P(S) \propto S^{\beta}$, with $0 < \beta <1$. We shall show that there is a functional dependence between the localization measure $A$ and the exponent $\beta$, namely that $\beta$ is a monotonic function of $A$: in the case of the strong localization are $A$ and $\beta$ small, while in the case of weak localization (almost extended chaotic states) $A$ and $\beta$ are close to $1$. We shall illustrate the approach in the model example of the above mentioned billiard family, where we can separate the regular and chaotic states. This presentation is based on our very recent papers.