# 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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