Statistical properties of the localization measure of chaotic eigenstates and the spectral statistics in a mixed-type billiard

Abstract: We study the quantum localization in the chaotic eigenstates of a billiard with mixed-type phase space, after separating the regular and chaotic eigenstates, in the regime of slightly distorted circle billiard where the classical transport time in the momentum space is still large enough, although the diffusion is not normal. In quantum systems with discrete energy spectrum the Heisenberg time $t_H =2\pi \hbar/\Delta E$, where $\Delta E$ is the mean level spacing, is an important time scale.The classical transport time scale $t_T$ in relation to the Heisenberg time scale $t_H$ (their ratio is the parameter $\alpha=t_H/t_T$) determines the degree of localization of the chaotic eigenstates, whose measure $A$ is based on the information entropy. We show that $A$ is linearly related to normalized inverse participation ratio. The localization of chaotic eigenstates is reflected also in the fractional power-law repulsion between the nearest energy levels in the sense that the probability density to find successive levels on a distance $S$ goes like $\propto S^\beta$ for small $S$, where $0\leq\beta\leq1$, and $\beta=1$ corresponds to completely extended states. We show that the level repulsion exponent $\beta$ is empirically a rational function of $\alpha$, and the mean $\langle A \rangle$ as a function of $\alpha$ is also well approximated by a rational function. In both cases there is some scattering of the empirical data around the mean curve, which is due to the fact that $A$ actually has a distribution, typically with quite complex structure, but in the limit $\alpha\rightarrow \infty$ well described by the beta distribution. Like in other systems, $\beta$ goes from $0$ to $1$ when $\alpha$ goes from $0$ to $\infty$. $\beta$ is a function of $\langle A \rangle$, similar to the quantum kicked rotator and the stadium billiard.