Generalized quantum mechanical two Coulomb centers problem (Demkov problem)

TL;DR

This paper introduces a new exactly solvable quantum two Coulomb centers problem with complex conjugate charges, expanding the class of solvable models and analyzing its spectral properties and boundary conditions.

Contribution

It presents a novel solvable model for the two Coulomb centers problem with complex parameters, including a detailed classification of boundary-value problems and spectral analysis.

Findings

New solvable quantum Coulomb problem with complex parameters

Classification of boundary-value problems for this system

Numerical insights into the energy spectrum structure

Abstract

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.