Spectral statistics and localization properties of a $C_3$-symmetric billiard

TL;DR

This paper investigates the spectral statistics and eigenstate localization of a C3-symmetric billiard, revealing GOE-GUE correspondence and quantum ergodicity onset through high-precision eigenvalue computations and entropy analysis.

Contribution

It introduces a high-precision eigenvalue computation method for symmetry subspaces, improving spectral analysis and confirming quantum ergodicity in a C3-symmetric billiard.

Findings

Clear GOE-GUE spectral correspondence observed.

Long-range spectral correlations are now well-resolved.

Eigenstate localization follows a Beta distribution with energy-dependent variance.

Abstract

We revisit the spectral statistics of the C$_3$--symmetric billiard introduced by Dembowski [Phys. Rev. E, R4516 (2000)], which exhibits both GOE and 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 2.8x10$^5$ 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 Beta distribution whose standard deviation decays as a power--law with energy, consistent with the onset of quantum ergodicity as described by Schnirelman's theorem.