Two electrons in an external oscillator potential: hidden algebraic structure

TL;DR

This paper reveals a hidden algebraic structure in the two-electron Coulomb problem within an oscillator potential, explaining existing solutions and uncovering degeneracies, marking a novel algebraic insight in atomic physics.

Contribution

It identifies a hidden $sl_2$-algebraic structure in a two-electron system, explaining known solutions and revealing degeneracies, a first in atomic physics.

Findings

Existence of hidden $sl_2$-algebraic structure in the problem

Explanation of known exact solutions

Discovery of degeneracies in energy levels

Abstract

It is shown that the Coulomb correlation problem for a system of two electrons (two charged particles) in an external oscillator potential possesses a hidden $sl_2$-algebraic structure being one of recently-discovered quasi-exactly-solvable problems. The origin of existing exact solutions to this problem, recently discovered by several authors, is explained. A degeneracy of energies in electron-electron and electron-positron correlation problems is found. It manifests the first appearence of hidden $sl_2$-algebraic structure in atomic physics.