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

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.Comment: LaTeX (v3): 10 pages, 9 figures (new figure 6 added), 1 table; final version for Phys. Rev. E (in print

Level density of the H\'enon-Heiles system above the critical barrier Energy

We discuss the coarse-grained level density of the H\'enon-Heiles system above the barrier energy, where the system is nearly chaotic. We use periodic orbit theory to approximate its oscillating part semiclassically via Gutzwiller's semiclassical trace formula (extended by uniform approximations for the contributions of bifurcating orbits). Including only a few stable and unstable orbits, we reproduce the quantum-mechanical density of states very accurately. We also present a perturbative calculation of the stabilities of two infinite series of orbits (R$_n$ and L$_m$), emanating from the shortest librating straight-line orbit (A) in a bifurcation cascade just below the barrier, which at the barrier have two common asymptotic Lyapunov exponents $\chi_{\rm R}$ and $\chi_{\rm L}$.Comment: LaTeX, style FBS (Few-Body Systems), 6pp. 2 Figures; invited talk at "Critical stability of few-body quantum systems", MPI-PKS Dresden, Oct. 17-21, 2005; corrected version: passages around eq. (6) and eqs. (12),(13) improve

Uniform approximations for pitchfork bifurcation sequences

In non-integrable Hamiltonian systems with mixed phase space and discrete symmetries, sequences of pitchfork bifurcations of periodic orbits pave the way from integrability to chaos. In extending the semiclassical trace formula for the spectral density, we develop a uniform approximation for the combined contribution of pitchfork bifurcation pairs. For a two-dimensional double-well potential and the familiar H\'enon-Heiles potential, we obtain very good agreement with exact quantum-mechanical calculations. We also consider the integrable limit of the scenario which corresponds to the bifurcation of a torus from an isolated periodic orbit. For the separable version of the H\'enon-Heiles system we give an analytical uniform trace formula, which also yields the correct harmonic-oscillator SU(2) limit at low energies, and obtain excellent agreement with the slightly coarse-grained quantum-mechanical density of states.Comment: LaTeX, 31 pp., 18 figs. Version (v3): correction of several misprint

Photoabsorption spectra of the diamagnetic hydrogen atom in the transition regime to chaos: Closed orbit theory with bifurcating orbits

With increasing energy the diamagnetic hydrogen atom undergoes a transition from regular to chaotic classical dynamics, and the closed orbits pass through various cascades of bifurcations. Closed orbit theory allows for the semiclassical calculation of photoabsorption spectra of the diamagnetic hydrogen atom. However, at the bifurcations the closed orbit contributions diverge. The singularities can be removed with the help of uniform semiclassical approximations which are constructed over a wide energy range for different types of codimension one and two catastrophes. Using the uniform approximations and applying the high-resolution harmonic inversion method we calculate fully resolved semiclassical photoabsorption spectra, i.e., individual eigenenergies and transition matrix elements at laboratory magnetic field strengths, and compare them with the results of exact quantum calculations.Comment: 26 pages, 9 figures, submitted to J. Phys.

Shell structure and orbit bifurcations in finite fermion systems

We first give an overview of the shell-correction method which was developed by V. M. Strutinsky as a practicable and efficient approximation to the general selfconsistent theory of finite fermion systems suggested by A. B. Migdal and collaborators. Then we present in more detail a semiclassical theory of shell effects, also developed by Strutinsky following original ideas of M. Gutzwiller. We emphasize, in particular, the influence of orbit bifurcations on shell structure. We first give a short overview of semiclassical trace formulae, which connect the shell oscillations of a quantum system with a sum over periodic orbits of the corresponding classical system, in what is usually called the "periodic orbit theory". We then present a case study in which the gross features of a typical double-humped nuclear fission barrier, including the effects of mass asymmetry, can be obtained in terms of the shortest periodic orbits of a cavity model with realistic deformations relevant for nuclear fission. Next we investigate shell structures in a spheroidal cavity model which is integrable and allows for far-going analytical computation. We show, in particular, how period-doubling bifurcations are closely connected to the existence of the so-called "superdeformed" energy minimum which corresponds to the fission isomer of actinide nuclei. Finally, we present a general class of radial power-law potentials which approximate well the shape of a Woods-Saxon potential in the bound region, give analytical trace formulae for it and discuss various limits (including the harmonic oscillator and the spherical box potentials).Comment: LaTeX, 67 pp., 30 figures; revised version (missing part at end of 3.1 implemented; order of references corrected