Quantum chaos in triangular billiards

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 π, whose classical dynamics is presumably mixing, and three with exactly one angle rational with π, which are presumably only weakly mixing or even nonergodic 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 a scarring or superscarring 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 analyzing distributions of phase-space localization measures of eigenstates and inspecting the structure of characteristic typical and atypical eigenfunctions.