Classical and quantum chaos in the generalized parabolic lemon-shaped billiard

Two-dimensional billiards of a generalized parabolic lemonlike shape are investigated classically and quantum mechanically depending on the shape parameter δ. Quantal spectra are analyzed by means of the nearest-neighbor spacing distribution method. Calculated results are well accounted for by the proposed new two-parameter distribution function $P\left(s\right),$ which is a generalization of Brody and Berry-Robnik distributions. Classically, Poincaré diagrams are shown and interpreted in terms of the lowest periodic orbits. For $\delta =2,$ the billiard has some unique characteristics resulting from the focusing property of the parabolic mirror. Comparison of the classical and quantal results shows an accordance with the Bohigas, Giannoni, and Schmit conjecture and confirms the relevance of the new distribution for the analysis of realistic spectral data.