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

We study the quantum localization in the chaotic eigenstates of a billiard with mixed-type phase space [J. Phys. A: Math. Gen. 16, 3971 (1983); 17, 1019 (1984)], 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. This is a continuation of our recent papers [Phys. Rev. E 88, 052913 (2013); 98, 022220 (2018)]. In quantum systems with discrete energy spectrum the Heisenberg time t(H) = 27 pi(h) over bar/Delta E, where Delta E is the mean level spacing (inverse energy level density), is an important timescale. The classical transport timescale t(T) (transport time) in relation to the Heisenberg timescale 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 (level spacing distribution) to find successive levels on a distance S goes like proportional to S-beta for small S, where 0 <= beta <= 1, and beta = 1 corresponds to completely extended states. We show that the level repulsion exponent beta is empirically alpha rational function of a, and the mean < A > (averaged over more than 1000 eigenstates) 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 -> infinity well described by the beta distribution. The scattering is significantly stronger than (but similar as) in the stadium billiard [Nonlin. Phenom. Complex Syst. (Minsk) 21, 225 (2018)] and the kicked rotator [Phys. Rev. E 91, 042904 (2015)]. Like in other systems, beta goes from 0 to 1 when alpha goes from 0 to infinity. beta is a function of < A >, similar to the quantum kicked rotator and the stadium billiard.