Quantum chaos in triangular billiards

TL;DR

This study numerically investigates spectral statistics and eigenfunctions of quantized triangular billiards, revealing that systems with generic irrational angles align with random matrix theory predictions, while others show deviations due to scarring effects.

Contribution

It provides the first extensive numerical analysis of spectral statistics in triangular billiards with various angles, extending quantum chaos conjectures to systems without hard chaos.

Findings

Generic irrational triangles match Gaussian orthogonal ensemble statistics.

Some cases show deviations attributed to scarring or super-scarring.

Eigenfunction analysis supports spectral statistics results.

Abstract

We present an extensive numerical study of spectral statistics and eigenfunctions of quantized triangular billiards. We compute two million consecutive eigenvalues for six representative cases of triangular billiards, three with generic angles with irrational ratios with $\pi$, whose classical dynamics is presumably mixing, and three with exactly one angle rational with $\pi$, which are presumably only weakly mixing or even only non-ergodic in case of right-triangles. We find excellent agreement of short and long range spectral statistics with the Gaussian orthogonal ensemble of random matrix theory for the most irrational generic triangle, while the other cases show small but significant deviations which are attributed either to scarring or super-scarring mechanism. This result, which extends the quantum chaos conjecture to systems with dynamical mixing in the absence of hard (Lyapunov) chaos, has been corroborated by analysing distributions of phase-space localisation measures of eigenstates and inspecting the structure of characteristic typical and atypical eigenfunctions.