Excited states of spherium

TL;DR

This paper presents analytic solutions for excited states of a two-electron model confined to a spherical surface, revealing polynomial solutions and discussing cusp conditions and degeneracies.

Contribution

It provides the first known analytic solutions for excited states in a Coulomb-interacting two-electron spherical model, expanding understanding of quasi-exact solvability.

Findings

Polynomial solutions for ground and some excited states (L ≤ 3)

Discussion of Kato cusp conditions and interdimensional degeneracies

Analytic expressions for energy levels and wavefunctions

Abstract

We report analytic solutions of a recently discovered quasi-exactly solvable model consisting of two electrons, interacting {\em via} a Coulomb potential, but restricted to remain on the surface of a $\mathcal{D}$-dimensional sphere. Polynomial solutions are found for the ground state, and for some higher ($L\le3$) states. Kato cusp conditions and interdimensional degeneracies are discussed.