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Normalization and global analysis of perturbations of the hydrogen atom

K. Efstathiou* and D. A. Sadovskií†

K. Efstathiou*

  • Department of Mathematics, University of Groningen, Groningen 9700 AK, The Netherlands

D. A. Sadovskií†

  • Département de Physique, Université du Littoral–Côte d’Opale, 59140 Dunkerque, France

  • *k.efstathiou@rug.nl
  • †sadovski@univ-littoral.fr
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Rev. Mod. Phys. 82, 2099 – Published 3 August, 2010

DOI: https://doi.org/10.1103/RevModPhys.82.2099

Abstract

The hydrogen atom perturbed by sufficiently small homogeneous static electric and magnetic fields of arbitrary mutual alignment is a specific perturbation of the Kepler system with three degrees of freedom and three parameters. Normalization of the Keplerian symmetry reveals that the parameter space is stratified into resonant zones of systems, each zone with an internal dynamical stratification of its own (Efstathiou, Sadovskií, and Zhilinskií, 2007, Proc. R. Soc. London, Ser. A 463, 1771). Based on the fully integrable approximation, the bundle of invariant tori of individual systems within zones is characterized globally and the qualitative dynamical stratification is uncovered. The techniques involved in this analysis are illustrated with the example of the 1:1 resonance zone (near orthogonal fields) whose structure is known at present. Applications in the corresponding quantum system are also described.

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