Solving a two-electron quantum dot model in terms of polynomial solutions of a Biconfluent Heun Equation

TL;DR

This paper derives exact solutions for a two-electron quantum dot model in two dimensions by transforming the Schrödinger equation into a biconfluent Heun equation, enabling analytical determination of part of the energy spectrum and eigenfunctions.

Contribution

It introduces a method to obtain exact polynomial solutions for the two-electron quantum dot problem using biconfluent Heun equations, including cases with magnetic fields.

Findings

Finite portion of energy spectrum determined analytically.

Eigenfunctions expressed in closed form.

Method applicable to magnetic field scenarios.

Abstract

The effects on the non-relativistic dynamics of a system compound by two electrons interacting by a Coulomb potential and with an external harmonic oscillator potential, confined to move in a two dimensional Euclidean space, are investigated. In particular, it is shown that it is possible to determine exactly and in a closed form a finite portion of the energy spectrum and the associated eigeinfunctions for the Schr\"odinger equation describing the relative motion of the electrons, by putting it into the form of a biconfluent Heun equation. In the same framework, another set of solutions of this type can be straightforwardly obtained for the case when the two electrons are submitted also to an external constant magnetic field.