Periodic orbit theory for the Hénon-Heiles system in the continuum region

Abstract

We investigate the resonance spectrum of the Hénon-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.05.45.Mt,03.65.Sq

Semiclassical trace formulas for pitchfork bifurcation sequences

“…It will then be an interesting question to study which periodic orbits are important in the continuum region. Work along these lines is in progress [36]. Although the continuum region is also classically unbounded, all the R and L orbits bifurcating from the A orbit (which itself ceases to exist above e = 1), as well as the D orbit bifurcating from C, continue to exist and are bounded at all energies e > 1 [18,34].…”

“…Here T u = T B = 2π, S u (E) = S B (E) = 2πE, and the period T A (E) and action S A (E) of the v motion are given in equations (34) - (36). The TF level density is then given by…”

“…An extension of the semiclassical analysis of the density of states to the energy region above the barrier (e > 1), including a rigorous quantum-mechanical determination of the widths and energy shifts due to barrier tunnelling in the quasi-bound region of the spectrum and of the resonances in the continuum region, is in progress [36].…”

Shell structure and orbit bifurcations in finite fermion systems

“…We note that the POT can also be applied to open systems above the continuum threshold, where the level density is dominated by resonances; an example has been given in [53]. As a first step towards dynamics, the oscillating parts of nuclear moments of inertia have been studied…”

“…where the level density is dominated by resonances; an example has been given in [53]. As a first step towards dynamics, the oscillating parts of nuclear moments of inertia have been studied semiclassically in terms of POs in [54,55].…”