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

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 in Phys. Today 46 38 (1993). We study two classical and quantum lemon billiards, for the cases B = 0.1953, 0.083, which are mixed-type billiards with complex structure of phase space, without significant stickiness regions. A preliminary study of their spectra was published recently (Physics 1 1-14 (2021)). We calculate a great number $10^6$ of consecutive eigenstates and their Poincar\'e-Husimi (PH) functions, and analyze their localization properties by studying the entropy localization measure and the normalized inverse participation ratio. We also 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 very good phenomenological overview of the structural and localization properties of PH functions in quantum mixed-type Hamiltonian systems.