Hydrogen Atom in de Sitter Spaces

The hydrogen atom theory is developed for the de Sitter and anti de Sitter spaces of constant negative curvature on the basis of the Klein-Gordon-Fock wave equation in static coordinates. In both models, after separation of variables, the problem is reduced to the general Heun equation, a second order linear differential equation having four regular singular points. Qualitative examination shows that the energy spectrum for the hydrogen atom in the de Sitter space should be quasi-stationary, and the atom should be unstable. We derive an approximate expression for energy levels within the quasi-classical approach and estimate the probability of decay of the atom. A similar analysis shows that in the anti de Sitter model the hydrogen atom should be stable in the quantum-mechanical sense. Using the quasi-classical approach, we derive approximate formulas for the energy levels for this case as well. Finally, we present the extension to the case of a spin 1/2 particle for both de Sitter models. This extension leads to complicated differential equations with 8 singular points