This is a review paper on up-to-date state of research in stationary quantum chaos, published in 2020 as a special chapter in a monography. We review the fundamental concepts of quantum chaos in Hamiltonian systems, as the basis to understand the properties of generic systems (so-called mixed-type systems, or systems with divided phase space), where the classical regular regions and chaotic regions coexist in the phase space. The quantum evolution of classically chaotic bound systems does not possess the sensitive dependence on initial conditions, and thus no chaotic behavior occurs, whereas the study of the stationary solutions of the Schrödinger equation in the quantum phase space (Wigner functions) reveals precise analogy of the structure of the classical phase portrait. We analyze the regular eigenstates associated with invariant tori in the classical phase space, and the chaotic eigenstates associated with the classically chaotic regions, and the corresponding energy spectra. Then we present the theoretical description of the generic (mixed-type) systems for the case of no quantum localization of the chaotic eigenstates (all states being extended) introduced by Berry and Robnik (J Phys A Math Gen 17:2413, 1984). The effects of quantum localization of the chaotic eigenstates set in when the Heisenberg time scale is shorter than the classical transport time (such as diffusion time), and these effects are treated phenomenologically, resulting in Brody-like level statistics for the chaotic states alone, which can be found also at very high-lying levels. Then we treat the level spacing distribution for the...
COBISS.SI-ID: 98261249
We study the aspects of quantum localization in the stadium billiard, which is a classically chaotic ergodic system, but in the regime of slightly distorted circle billiard the diffusion in the momentum space is very slow. In quantum systems with discrete energy spectrum the Heisenberg time tH = 2 pi=delta E, where deltaE is the mean level spacing (inverse energy level density), is an important time scale. The classical transport time scale tT (diffusion time) in relation to the Heisenberg time scale tH (their ratio is the parameter alfa = tH=tT ) determines the degree of localization of the chaotic eigenstates, whose measure A is based on the information entropy. 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 alfa S beta for small S, where 0 equal or less 1, and beta = 1 corresponds to completely extended states. We show that the level repulsion exponent beta is a unique rational function of alfa, and A is a unique rational function of alfa. Beta goes from 0 to 1 when alfa goes from 0 to endless. Also, beta is a linear function of A, which is similar as in the quantum kicked rotator, but different from a mixed type billiard.
COBISS.SI-ID: 95735297
We study statistical properties of the localization measure and the spectral statistics in a system with a mixed-type phase space, so-called Robnik billiard. We calculate and extensively analyze the Husimi functions of the eigenstates, separate the regular and chaotic eigenstates, calculate the localization measure A of the chaotic eigenstates, based on the information entropy, and show that it is equivalent (proportional) to the normalized inverse participation ratio. A is distributed according to the beta distribution in the case of uniformly chaotic chaotic eigenstates, while its distribution is nonuniversal in cases of cantori and stickiness in the classical phase space. The Brody level repulsion parameter of the chaotic eigenstates is approximately linearly proportional to the mean value of A, and is a rational function of alfa, which is the ratio of the Heisenberg time and classical transport time.
COBISS.SI-ID: 98013185