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The boundary of the lemon billiards is defined by the intersection of twocircles of equal unit radius with the distance 2B between their centers, asintroduced by Heller and Tomsovic in Phys. Today 46 38 (1993). We study twoclassical and quantum lemon billiards, for the cases B = 0.1953, 0.083, whichare mixed-type billiards with complex structure of phase space, withoutsignificant stickiness regions. A preliminary study of their spectra waspublished recently (Physics 1 1-14 (2021)). We calculate a great number $10^6$of consecutive eigenstates and their Poincar\'e-Husimi (PH) functions, andanalyze their localization properties by studying the entropy localizationmeasure and the normalized inverse participation ratio. We also introduce anoverlap index which measures the degree of the overlap of PH functions withclassically regular and chaotic regions. We observe the existence of regularstates associated with invariant tori and chaotic states associated with theclassically chaotic regions, and also the mixed-type states. We show that inaccordance with the Berry-Robnik picture and the principle of uniformsemiclassical condensation of PH functions the relative fraction of mixed-typestates decreases as a power law with increasing energy, thus in the strictsemiclassical limit leaving only purely regular and chaotic states. Ourapproach offers a general very good phenomenological overview of the structuraland localization properties of PH functions in quantum mixed-type Hamiltoniansystems.

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