Generalized quantum mechanical two Coulomb centers problem (Demkov problem)

We present a new exactly solvable quantum problem for which the Schroedinger equation allows for separation of variables in oblate spheroidal coordinates. Namely, this is the quantum mechanical two Coulomb centers problem for the case of imaginary intercenter parameter and complex conjugate charges is considered. Since the potential is defined by the two-sheeted mapping whose singularities are concentrated on a circle rather than at separate points, there arise additional possibilities in choice of boundary conditions. Detailed classification of the various types of boundary-value problems is given. The quasi-radial equation leads to a new type of boundary value problems which was never considered before. Results of the numerical calculations allowing to draw conclusions about the structure of the energy spectrum are shown. Possible physical applications are discussed.