Elementary solutions of the quantum planar two center problem

TL;DR

This paper derives simple analytical solutions for a quantum electron in a plane influenced by two Coulomb centers at specific distances, using quasi-exact solvability and Heun equations, highlighting differences when centers have equal strength.

Contribution

It provides explicit elementary solutions for the planar two-center quantum problem at special distances, extending previous three-dimensional results and analyzing the equal-strength case.

Findings

Elementary solutions exist at specific inter-center distances.

Solutions are derived using quasi-exact solvability and Heun equations.

No elementary solutions for equal-strength centers.

Abstract

The quantum problem of an electron moving in a plane under the field created by two Coulombian centers admits simple analytical solutions for some particular inter-center distances. These elementary eigenfunctions, akin to those found by Demkov for the analogous three dimensional problem, are calculated using the framework of quasi-exact solvability of a pair of entangled ODE's descendants from the Heun equation. A different but interesting situation arises when the two centers have the same strength. In this case completely elementary solutions do not exist.