Universal signatures of quantum chaos

Phenomenology of quantum eigenstates in mixed-type systems: Lemon billiards with complex phase space structure

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)0031-922810.1063/1.881358]. This paper is a continuation of our recent papers on a classical and quantum ergodic lemon billiard (B=0.5) with strong stickiness effects [Č. Lozej et al., Phys. Rev. E 103, 012204 (2021)2470-004510.1103/PhysRevE.103.012204], as well as on the three billiards with a simple mixed-type phase space and no stickiness [Č. Lozej et al., Nonlin. Phenom. Complex Syst. 24, 1 (2021)1817-245810.33581/1561-4085-2021-24-1-1-18]. 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 [ Č. Lozej, D. Lukman, and M. Robnik, Physics 3, 888 (2021)10.3390/physics3040055]. We calculate a very large number (10^{6}) 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.

Periodic orbit theory and the statistical analysis of scaling quantum graph spectra

The explicit solution to the spectral problem of quantum graphs found recently by Dabaghian and Blümel [Phys. Rev. E 68, 055201(R) (2003); 70, 046206 (2004); JETP Lett. 77, 530 (2003)] is used to produce an exact periodic orbit theory description for the probability distributions of spectral statistics, including the distribution for the nearest neighbor separations sn = kn - kn-1, and the distribution of the spectral oscillations around the average, deltakn=kn - kn.

Level spacings and periodic orbits

Starting from a semiclassical quantization condition based on the trace formula, we derive a periodic-orbit formula for the distribution of spacings of eigenvalues with k intermediate levels. Numerical tests verify the validity of this representation for the nearest-neighbor level spacing (k=0). In a second part, we present an asymptotic evaluation for large spacings, where consistency with random matrix theory is achieved for large k. We also discuss the relation with the method of Bogomolny and Keating [Phys. Rev. Lett. 77, 1472 (1996)] for two-point correlations.