Periodic orbit theory for the H\'enon-Heiles system in the continuum region

Abstract

We investigate the resonance spectrum of the H\'enon-Heiles potential up to twice the barrier energy. The quantum spectrum is obtained by the method of complex coordinate rotation. We use periodic orbit theory to approximate the oscillating part of the resonance spectrum semiclassically and Strutinsky smoothing to obtain its smooth part. Although the system in this energy range is almost chaotic, it still contains stable periodic orbits. Using Gutzwiller's trace formula, complemented by a uniform approximation for a codimension-two bifurcation scenario, we are able to reproduce the coarse-grained quantum-mechanical density of states very accurately, including only a few stable and unstable orbits.