Arithmetic and pseudo-arithmetic billiards

TL;DR

This paper investigates the spectral properties of arithmetic triangular billiards, revealing how parity-based subspectra influence quantum level statistics and distinguishing genuinely arithmetic from pseudo-arithmetic cases.

Contribution

It introduces a parity-based subdivision of length spectra and clarifies conditions for Poissonian versus GOE level statistics in arithmetic billiards.

Findings

Length spectra divide into parity-based subspectra.

Genuinely arithmetic billiards exhibit Poissonian statistics.

Pseudo-arithmetic billiards follow GOE universality class.

Abstract

The arithmetic triangular billiards are classically chaotic but have Poissonian energy level statistics, in ostensible violation of the BGS conjecture. We show that the length spectra of their periodic orbits divides into subspectra differing by the parity of the number of reflections from the triangle sides; in the quantum treatment that parity defines the reflection phase of the orbit contribution to the Gutzwiller formula for the energy level density. We apply these results to all 85 arithmetic triangles and establish the boundary conditions under which the quantum billiard is \textquotedblleft genuinely arithmetic\textquotedblright, i. e., has Poissonian level statistics; otherwise the billiard is "pseudo-arithmetic" and belongs to the GOE universality class