  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.
