Two-Dimensional Supersymmetric Quantum Mechanics: Two Fixed Centers of Force

TL;DR

This paper explores two methods for constructing supersymmetric quantum mechanics models involving two Coulombian centers, analyzing their spectral problems and solutions, including special cases where equations simplify.

Contribution

It introduces two approaches to supersymmetrize the two-center Coulomb problem and analyzes their spectral equations, including reductions to well-known special functions.

Findings

Spectral problems relate to Razavy and Whittaker-Hill equations.

When centers have equal strength, equations reduce to Mathieu equations.

Zero-energy ground states can be found even in complex cases.

Abstract

The problem of building supersymmetry in the quantum mechanics of two Coulombian centers of force is analyzed. It is shown that there are essentially two ways of proceeding. The spectral problems of the SUSY (scalar) Hamiltonians are quite similar and become tantamount to solving entangled families of Razavy and Whittaker-Hill equations in the first approach. When the two centers have the same strength, the Whittaker-Hill equations reduce to Mathieu equations. In the second approach, the spectral problems are much more difficult to solve but one can still find the zero-energy ground states.