Research plan to apply for Humboldt Grant, October 1997

Resonances and Rydberg states of atoms in external fields via anharmonic-oscillator semiclassical perturbation theory with complex coefficients

1. Introduction

Effects of external electric and magnetic fields on highly excited atoms have attracted both experimental and theoretical interest, see, for example, the review [1]. For an electric field only, this problem can be easily solved numerically because it is separable (in parabolic coordinates). Several analytical methods were also proposed, see, for example [2], where modifications of the semiclassical WKB method and a perturbation theory were used both for low-lying states and for over the barrier Stark resonances. In the presence of a magnetic field this problem is harder to solve because it is no longer separable. Analytic solutions exist only in the limit of small fields when perturbation theory is applicable. Computations for intermediate and strong fields are done usually by expanding the wave function over a large basis set.

The subject of this research is the development of new advanced methods of perturbation theory that can predict and qualitatively describe new classes of atomic phenomena like resonances over the ionization threshold. Perturbation theory is a usual tool for the ground and lower excited states in weak fields. However, the applied fields can dominate atomic forces for highly excited states, and the resulting spectrum cannot be understood within the framework of traditional perturbation theory (in powers of the field strength). The main idea of the present approach is to use the perturbation theory that is similar to a semiclassical expansion in the theory of molecular vibrations rather than to use a traditional expansion for the perturbed hydrogen atom. Such an approach may become useful in a much broader context, i.e. wherever saddle-point resonances occur [3].

Generally, the results may be useful for astrophysics for detailed analyses of spectra of atoms in interstellar magnetic fields as well as for laboratory studies of highly excited Rydberg atoms.

The general approach to be used here will be described briefly.

2. Semiclassical expansion associated with a potential minimum or complex stationary points

Consider, for example, any quantum Hamiltonian of the form . . .

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