Spectral statistics and localization properties of a C_{3}-symmetric billiard

We revisit the spectral statistics of the C_{3}-symmetric billiard introduced by F. Leyvraz et al. [J. Phys. A: Math. Gen. 29, L575 (1996)0305-447010.1088/0305-4470/29/22/004], which exhibits both Gaussian orthogonal ensemble (GOE) and Gaussian unitary ensemble (GUE) statistics depending on the symmetry block. Using high-precision Beyn's contour-integral method for the nonlinear Fredholm eigenvalue problem with built-in separation of irreducible subspaces, we compute 2.8×10^{5} eigenvalues in each symmetry subspace, enabling statistically meaningful comparisons with random matrix theory. The improved spectra reveal clear GOE-GUE correspondence and resolve previously observed deviations in long-range spectral correlations. Furthermore, we analyze phase-space eigenstate localization through the distribution of entropy localization measures, which, for chaotic states follow a β distribution whose standard deviation decays as a power law with energy, consistent with the onset of quantum ergodicity as described by Schnirelman's theorem.