Generalized quantum mechanical two-Coulomb-center problem (Demkov problem)

Abstract

We present a new exactly solvable quantum problem for which the Schrödinger 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. PACS number(s): 03.65. Ge , 12.39.Pn , 31.90 + s

New representations for square-integrable spheroidal functions

“…In works [1][2][3][4][5] various quantum potential models that allow separation of variables in the Schrödinger equation ∆Ψ + 2 (E − V ) Ψ = 0 (1) in spheroidal coordinates were considered. Particular attention was drawn to the problems with separation of variables in oblate spheroidal coordinates due to the specificity of the emerging boundary value problems.…”

“…where, the values of k and m are integer parameters, and k is the number of zeros of the eigenfunction X mk (ξ) on the interval (−∞, +∞). This problem first arose in work [1] in connection with the study of the discrete spectrum of the generalized quantum mechanical problem of two centers; therefore, a square integrability was additionally required:…”