  The hydrogen atom theory is developed for the de Sitter and anti de Sitter
spaces on the basis of the Klein-Gordon-Fock wave equation in static
coordinates. In both models, after separation of the variables, the problem is
reduced to the general Heun equation, a second order linear differential
equation having four regular singular points. A 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
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.
