Semiclassical treatment of symmetry breaking and bifurcations in a non-integrable potential

TL;DR

This paper develops an analytical semiclassical trace formula for the Hénon-Heiles potential, accurately capturing symmetry breaking and bifurcations, and aligns well with quantum results for energy level densities.

Contribution

It introduces an improved stationary phase method to extend Gutzwiller's trace formula, effectively handling symmetry-breaking points in a non-integrable potential.

Findings

Good agreement with quantum level densities

Accurate description of symmetry breaking effects

Unified treatment of critical points in semiclassical analysis

Abstract

We have derived an analytical trace formula for the level density of the H\'enon-Heiles potential using the improved stationary phase method, based on extensions of Gutzwiller's semiclassical path integral approach. This trace formula has the correct limit to the standard Gutzwiller trace formula for the isolated periodic orbits far from all (critical) symmetry-breaking points. It continuously joins all critical points at which an enhancement of the semiclassical amplitudes occurs. We found a good agreement between the semi- classical and the quantum oscillating level densities for the gross shell structures and for the energy shell corrections, solving the symmetry breaking problem at small energies.