Anomalous shell effect in the transition from a circular to a triangular billiard

TL;DR

This paper investigates how a shell effect persists and manifests during the smooth transition of a billiard shape from circular to triangular, revealing a bifurcation-driven origin of the effect through semiclassical analysis.

Contribution

It demonstrates that a pronounced shell effect occurs during the shape transition, explained by a codimension-two bifurcation of the triangular orbit, using semiclassical trace formulas.

Findings

Shell effect remains strong during shape transition.

Bifurcation of the triangular orbit explains the shell effect.

Semiclassical trace formula accurately describes quantum level density.

Abstract

We apply periodic orbit theory to a two-dimensional non-integrable billiard system whose boundary is varied smoothly from a circular to an equilateral triangular shape. Although the classical dynamics becomes chaotic with increasing triangular deformation, it exhibits an astonishingly pronounced shell effect on its way through the shape transition. A semiclassical analysis reveals that this shell effect emerges from a codimension-two bifurcation of the triangular periodic orbit. Gutzwiller's semiclassical trace formula, using a global uniform approximation for the bifurcation of the triangular orbit and including the contributions of the other isolated orbits, describes very well the coarse-grained quantum-mechanical level density of this system. We also discuss the role of discrete symmetry for the large shell effect obtained here.