Level statistics in arithmetical and pseudo-arithmetical chaos

TL;DR

This paper explains why certain fully chaotic billiards with degenerate periodic orbits exhibit either Poissonian or Wigner-Dyson spectral statistics, depending on boundary conditions, by analyzing Maslov indices and phase contributions.

Contribution

It provides a semiclassical explanation for the spectral statistics in pseudo-arithmetical chaos, extending understanding of quantum chaos in systems with degenerate orbits.

Findings

Degenerate periodic orbits lead to either Poisson or Wigner-Dyson statistics.

Maslov indices determine phase coherence and spectral correlations.

The semiclassical approach explains spectral universality in pseudo-arithmetical systems.

Abstract

We resolve a long-standing riddle in quantum chaos, posed by certain fully chaotic billiards with constant negative curvature whose periodic orbits are highly degenerate in length. Depending on the boundary conditions for the quantum wave functions, the energy spectra either have uncorrelated levels usually associated with classical integrability or conform to the "universal" Wigner-Dyson type although the classical dynamics in both cases is the same. The resolution turns out surprisingly simple. The Maslov indices of orbits within multiplets of degenerate length either yield equal phases for the respective Feynman amplitudes (and thus Poissonian level statistics) or give rise to amplitudes with uncorrelated phases (leading to Wigner-Dyson level correlations). The recent semiclassical explanation of spectral universality in quantum chaos is thus extended to the latter case of "pseudo-arithmetical" chaos.