Effects of stickiness in the classical and quantum ergodic lemon billiard

We study the classical and quantum ergodic lemon billiard introduced by Heller Tomsovic in Phys. Today 46(7), 38 (1993), for case $B=1/2$, which is a classically system (without rigorous proof) exhibiting strong stickiness regions around zero-measure bouncing ball modes. The structure of uncovered S-plots Lozej [Phys. Rev. E 101, 052204 (2020)]. A unique transport or diffusion time cannot be defined. As consequence states are characterized following nonuniversal properties: (i) All eigenstates chaotic but localized as exhibited Poincar\'e-Husimi (PH) functions. (ii) entropy localization measure (also normalized inverse participation ratio) has distribution, typically bimodal, thus deviating from beta latter one being characteristic uniformly systems with no regions. (iii) energy-level spacing distribution Berry-Robnik-Brody (BRB), capturing two effects: quantally divided phase space (because most PH functions either inner-ones outer-ones, dictated stickiness, an effective parameter $\mu_1$ measuring size inner region bordered sticky invariant object, namely, cantorus), level repulsion (Brody) $\beta$. (iv) In energy range considered (between 20 000 to 400 above ground state) picture (the statistics spectra) not changing qualitatively, $\beta$ fluctuates 0.8, while decreases almost monotonically, increasing energy.