Elliptic billiard - a non-trivial integrable system

TL;DR

This paper studies the quantum elliptic billiard, revealing its integrable nature, unique spectral properties, and the significant role of non-isolated periodic orbits in its spectral fluctuations.

Contribution

It provides a detailed semiclassical analysis of the elliptic billiard, highlighting novel long-range spectral oscillations and the impact of periodic orbits on spectral statistics.

Findings

Elliptic billiard is confirmed as a generic integrable system.

Spectral analysis shows long-range oscillations in second order statistics.

Periodic orbits, especially non-isolated ones, dominate spectral fluctuations.

Abstract

We investigate the semiclassical energy spectrum of quantum elliptic billiard. The nearest neighbor spacing distribution, level number variance and spectral rigidity support the notion that the elliptic billiard is a generic integrable system. However, second order statistics exhibit a novel property of long-range oscillations. Classical simulation shows that all the periodic orbits except two are not isolated. In Fourier analysis of the spectrum, all the peaks correspond to periodic orbits. The two isolated periodic orbits have small contribution to the fluctuation of level density, while non-isolated periodic orbits have the main contribution. The heights of the majority of the peaks match our semiclassical theory except for type-O periodic orbits. Elliptic billiard is a nontrivial integrable system that will enrich our understanding of integrable systems.