Eur. Phys. J. D 25, 129-138 (2003)
https://doi.org/10.1140/epjd/e2003-00238-x

Uniform semiclassical approximations on a topologically non-trivial configuration space

The hydrogen atom in an electric field

Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany

Corresponding author: ^a bartsch@theo1.physik.uni-stuttgart.de

Received: 19 February 2003
Published online: 30 July 2003

Abstract

Semiclassical periodic-orbit theory and closed-orbit theory represent a quantum spectrum as a superposition of contributions from individual classical orbits. Close to a bifurcation, these contributions diverge and have to be replaced with a uniform approximation. Its construction requires a normal form that provides a local description of the bifurcation scenario. Usually, the normal form is constructed in flat space. We present an example taken from the hydrogen atom in an electric field where the normal form must be chosen to be defined on a sphere instead of a Euclidean plane. In the example, the necessity to base the normal form on a topologically non-trivial configuration space reveals a subtle interplay between local and global aspects of the phase space structure. We show that a uniform approximation for a bifurcation scenario with non-trivial topology can be constructed using the established uniformization techniques. Semiclassical photo-absorption spectra of the hydrogen atom in an electric field are significantly improved when based on the extended uniform approximations.