Tuning the hydrogen atom in crossed fields between the Zeeman and Stark limits

We consider the hydrogen atom in the orthogonal electric and magnetic fields whose strength is assumed to be small enough for the Coulomb n-shell perturbation theory to apply. Appropriate scaling of the two fields leads to a uniform parametrization of the problem by S, the combined strength of the two fields, and α, the ratio of the two field strengths. The initial six-dimensional phase space R₆ is lifted to the standard Kustaanheimo-Stiefel eight-dimensional space and then reduced explicitly to the S₂×S₂ reduced space of the n shell using the Lie transformation to the third order in S. At fixed S the system is uniformly tuned between the Zeeman and the Stark limits using the analytic formulas of the perturbation theory. The approach requires application of the invariant theory, group theory, and topology to the analysis of the dynamics on the reduced space S₂×S₂ and subsequent explicit transition to the original R₆. In particular we follow the evolution of the four principal periodic orbits (nonlinear normal modes) and corresponding four relative equilibria on S₂×S₂.