# Most typical 1:2 resonant perturbation of the hydrogen atom by weak electric and magnetic fields

### Abstract

We study a perturbation of the hydrogen atom by small homogeneous static electric and magnetic fields in a specific mutual alignment with angle approximately $π/3$ which results in the 1:2 resonance of the linearized Keplerian n-shell approximation. The stratified image of the energy-momentum map (bifurcation diagram) of the classical integrable approximation has for most such field configurations the same typical structure which we describe. The structure of the corresponding quantum energy spectrum, which we describe in detail, is in certain ways an analogue of the well known degeneracy found by Herrick [Phys. Rev. A 26, 323 (1982)] for the quadratic Zeeman effect and present also for many near orthogonal field configurations.