Effects of stickiness in quantum chaotic billiards with n-fold symmetry

In this work, we study a family of fully chaotic billiards that exhibits only rotational symmetries, whose geometry is based on the C₃ symmetry system proposed by Leyvraz, Schmit, and Seligman (LSS) in 1996. Quantum spectral analyses are performed on billiards with symmetry C_n (the billiard repeats itself under rotations of 2π/n), where n is the symmetry parameter. In these systems, there are subspaces of singlets (invariant under time reversal) and doublets (not invariant under time reversal). For the LSS billiard, it has been established both numerically and experimentally that the corresponding subspectra follow the Gaussian Orthogonal Ensemble (GOE) statistics for singlets and the Gaussian Unitary Ensemble (GUE) statistics for doublets. From a classical perspective, the shapes of these billiards allow certain subregions of phase space to be visited more frequently by chaotic trajectories, a phenomenon known as stickiness. We investigate the relationship between the fraction of sticky regions in classical phase space and the deviations of the energy subspectra from GOE and GUE statistics. Our results suggest the existence of correlations between the energy distributions associated with different symmetry subspaces. In addition, we discuss aspects related to the superposition of the different energy subspectra.