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Fluctuating number of energy levels in mixed-type lemon billiards
Črt Lozej, Dragan Lukman, Marko Robnik, 2021, original scientific article

Abstract: In this paper, the fluctuation properties of the number of energy levels (mode fluctuation) are studied in the mixed-type lemon billiards at high lying energies. The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between the centers, as introduced by Heller and Tomsovic. In this paper, the case of two billiards, defined by B = 0.1953, 0.083, is studied. It is shown that the fluctuation of the number of energy levels follows the Gaussian distribution quite accurately, even though the relative fraction of the chaotic part of the phase space is only 0.28 and 0.16, respectively. The theoretical description of spectral fluctuations in the Berry-Robnik picture is discussed. Also, the (golden mean) integrable rectangular billiard is studied and an almost Gaussian distribution is obtained, in contrast to theory expectations. However, the variance as a function of energy, E, behaves as - E, in agreement with the theoretical prediction by Steiner.
Keywords: nonlinear dynamics, quantum chaos, mixed-type systems, energy level statistics, lemon billiards, billiards
Published in DKUM: 13.10.2023; Views: 518; Downloads: 19
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Phenomenology of quantum eigenstates in mixed-type systems: Lemon billiards with complex phase space structure
Črt Lozej, Dragan Lukman, Marko Robnik, 2022, original scientific article

Abstract: The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between their centers, as introduced by Heller and Tomsovic [E. J. Heller and S. Tomsovic, Phys. Today 46, 38 (1993)]. This paper is a continuation of our recent papers on a classical and quantum ergodic lemon billiard (B = 0.5) with strong stickiness effects [C. Lozej ˇ et al., Phys. Rev. E 103, 012204 (2021)], as well as on the three billiards with a simple mixed-type phase space and no stickiness [C. Lozej ˇ et al., Nonlin. Phenom. Complex Syst. 24, 1 (2021)]. Here we study two classical and quantum lemon billiards, for the cases B = 0.1953, 0.083, which are mixed-type billiards with a complex structure of phase space, without significant stickiness regions. A preliminary study of their spectra was published recently [ C. Lozej, D. Lukman, and M. ˇ Robnik, Physics 3, 888 (2021)]. We calculate a very large number (106) of consecutive eigenstates and their Poincaré-Husimi (PH) functions, and analyze their localization properties by studying the entropy localization measure and the normalized inverse participation ratio. We introduce an overlap index, which measures the degree of the overlap of PH functions with classically regular and chaotic regions. We observe the existence of regular states associated with invariant tori and chaotic states associated with the classically chaotic regions, and also the mixed-type states. We show that in accordance with the Berry-Robnik picture and the principle of uniform semiclassical condensation of PH functions, the relative fraction of mixed-type states decreases as a power law with increasing energy, thus, in the strict semiclassical limit, leaving only purely regular and chaotic states. Our approach offers a general phenomenological overview of the structural and localization properties of PH functions in quantum mixed-type Hamiltonian systems.
Keywords: quantum physics, energy, localization, quantum chaos, billiards, chaotic systems
Published in DKUM: 12.10.2023; Views: 288; Downloads: 19
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