Quantum-classical correspondences of the Berry-Robnik parameter through bifurcations in lemon billiard systems

The quantum level statistics affected by bifurcations in classical dynamics is studied by using a one-parameter family of lemon billiard systems. The classical phase space of our system consists of regular and irregular regions. We determine an analytic solution of the phase volume for these regions as a function of the system parameter and show that the function reveals a cusp singularity at the bifurcation point. The function is compared with its quantum mechanical counterpart, the Berry-Robnik parameter. By estimating the semiclassical regime from the effective Planck constant that validates the quantum-classical correspondence of the Berry-Robnik parameter, we determine a region of the system parameter where the cusp can be reproduced by the statistical properties of the eigenenergy levels.