In this work we study the level spacing distribution in the classically mixed-type quantum systems (which are generic), exhibiting regular motion on invariant tori for some initial conditions and chaotic motion for the complementary initial conditions. In the asymptotic regime of the sufficiently deep semiclassical limit (sufficiently small effective Planck constant) the Berry and Robnik (1984 J. Phys. A: Math. Gen. 17 2413) picture applies, which is very well established. We present a new quasi-universal semiempirical theory of the level spacing distribution in a regime away from the Berry-Robnik regime (the near semiclassical limit), by describing both the dynamical localization effects of chaotic eigenstates, and the tunneling effects which couple regular and chaotic eigenstates. The theory works extremely well in the 2D mixed-type billiard system introduced by Robnik (1983 J. Phys. A: Math. Gen. 16 3971) and is also tested in other systems (mushroom billiard and Prosen billiard).

The phenomenon of quantum localization in classically chaotic eigenstates is one of the main issues in quantum chaos (or wave chaos), and thus plays an important role in general quantum mechanics or even in general wave mechanics. In this work we propose two different localization measures characterizing the degree of quantum localization, and study their relation to another fundamental aspect of quantum chaos, namely the (energy) spectral statistics. Our approach and method is quite general, and we apply it to billiard systems. One of the signatures of the localization of chaotic eigenstates is a 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 [proportionality]S(β) for small S, where 0≤β≤1, and β=1 corresponds to completely extended states. We show that there is a clear functional relation between the exponent β and the two different localization measures. One is based on the information entropy and the other one on the correlation properties of the Husimi functions. We show that the two definitions are surprisingly linearly equivalent. The approach is applied in the case of a mixed-type billiard system [M. Robnik, J. Phys. A: Math. Gen. 16, 3971 (1983)], in which the separation of regular and chaotic eigenstates is performed.

We study the quantum mechanics of a billiard (Robnik 1983) in the regime of mixed-type classical phase space (the shape parameter λ = 0.15) at very high-lying eigenstates, starting at about 1.000.000th eigenstate and including the consecutive 587654 eigenstates. By calculating the normalized Poincaré Husimi functions of the eigenstates and comparing them with the classical phase space structure, we introduce the overlap criterion which enables us to separate with great accuracy and reliability the regular and chaotic eigenstates, and the corresponding energies. The chaotic eigenstates appear all to be dynamically localized, meaning that they do not occupy uniformly the entire available chaotic classical phase space component, but are localized on a proper subset. We find with unprecedented precision and statistical significance that the level spacing distribution of the regular levels obeys the Poisson statistics, and the chaotic ones obey the Brody statistics, as anticipated in a recent paper by Batistić and Robnik (2010), where the entire spectrum was found to obey the BRB statistics. There are no effects of dynamical tunneling in this regime, due to the high energies, as they decay exponentially with the inverse effective Planck constant which is proportional to the square root of the energy.

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, 1049 (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 tH = 2π /∆E, where ∆E is the mean level spacing (inverse energy level density), is an important time scale. The classical transport time scale tT (transport time) in relation to the Heisenberg time scale tH (their ratio is the parameter α = tH /tT ) 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 ∝ S β for small S, where 0 ≤ β ≤ 1, and β = 1 corresponds to completely extended states. We show that the level repulsion exponent β is empirically a rational function of α, and the mean A (averaged over more than 1000 eigenstates) as a function of α 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 α → ∞ well described by the beta distribution. The scattering is significantly stronger than (but similar as) in the stadium billiard (Nonl.Phen.Compl.Sys. 21, No3, 225 (2018)) and the kicked rotator (Phys.Rev. E 91, 042904 ( 2015)). Like in other systems, β goes from 0 to 1 when α goes from 0 to ∞. β is a function of A , similar to the quantum kicked rotator and the stadium billiard.

We demonstrate that the energy or quasienergy level spacing distribution in dynamically localized chaotic eigenstates is excellently described by the Brody distribution, displaying the fractional power law level repulsion. This we show in two paradigmatic systems, namely for the fully chaotic eigenstates of the kicked rotator at K = 7, and for the chaotic eigenstates in the mixed-type billiard system (Robnik 1983), after separating the regular and chaotic eigenstates by means of the Poincaré Husimi function, at very high energies with great statistical significance (587654 eigenstates, starting at about 1.000.000 above the ground state). This separation confirms the Berry-Robnik picture, and is performed for the first time at such high energies.Introduction. -One of the main findings in quantum chaos [1-3] of stationary Schrödinger equation is the fact that the statistics of spectral fluctuations of the discrete quantal energy spectra around the smooth mean behaviour of classically chaotic systems obeys the Random Matrix Theory (RMT), in terms of the Gaussian ensembles of random matrices [1,2,4,5], in the sufficiently deep (or far) semiclassical limit. This finding is known as the Bohigas -Giannoni -Schmit (BGS) conjecture first published in [6], although some preliminary ideas were introduced in [7]. By "sufficiently deep (or far) semiclassical limit" we mean that some semiclassical condition is satisfied, namely that all relevant classical transport times, like the typical ergodic time, or diffusion time, are smaller than the so-called Heisenberg time, or break time, given by t H = 2π /∆E, where h = 2π is the Planck constant and ∆E is the mean energy level spacing, such that the mean energy level density is ρ(E) = 1/∆E. If the stated condition is satisfied, the quantum eigenstates as represented in the "quantum phase space" by the Wigner functions, or Husimi functions, are uniformly extended [3], and the spectral statistics is as in RMT, namely like for Gaussian Orthogonal Ensemble (GOE) or Gaussian Unitary Ensemble (GUE), depending on the antiunitary symmetries of the system [1-4, 8, 9]. Here we treat only the