Bifurcation and anomalous spectral accumulation in oval billiard

TL;DR

This paper investigates how bifurcations in the classical dynamics of an oval billiard influence quantum spectral statistics, revealing anomalous spectral accumulation and oscillations linked to classical bifurcating orbits.

Contribution

It demonstrates the connection between classical bifurcations and quantum spectral anomalies, providing numerical evidence and semiclassical analysis of spectral oscillations.

Findings

Eigenenergy levels show anomalous accumulation at bifurcation points.

Oscillations in the two-point correlation function are linked to bifurcating orbits.

Period of spectral oscillations matches the classical bifurcation orbit period.

Abstract

Spectral statistics of quantum oval billiard whose classical dynamical system shows bifurcations is numerically investigated in terms of the two-point correlation function (TPCF) which is defined as the probability density of finding two levels at a specific energy interval. The eigenenergy levels at bifurcation point is found to show anomalous accumulation which is observed as a periodic spike oscillation of the TPCF. We analyzed the eigenfunctions localizing onto the various classical trajectories in the phase space and found that the oscillation is supplied from a limited region in the phase space, which contains the bifurcating orbit. We also show that the period of the oscillation is in good agreement with the period of a contribution from the bifurcating orbit to the semiclassical TPCF obtained by Gutzwiller trace formula.