Two-body Coulomb problem and $g^{(2)}$ algebra (once again about the Hydrogen atom)

TL;DR

This paper explores the hidden algebraic structures in the hydrogen atom and related Coulomb problems, revealing new orthogonal polynomials and eigenfunctions linked to $g^{(2)}$ algebra and integrable systems.

Contribution

It introduces new polynomial eigenfunctions in Coulomb problems related to the $g^{(2)}$ algebra and connects them to the Zeeman effect and integrable three-body systems.

Findings

New orthogonal polynomials in two variables $(r, ho^2)$

Eigenfunctions related to $g^{(2)}$ algebra

Connection to Zeeman effect and integrable systems

Abstract

Taking the Hydrogen atom as an example it is shown that if the symmetry of a three-dimensional system is $O(2) \oplus Z_2$, the variables $(r, \rho, \varphi)$ allow a separation of the variable $\varphi$, and the eigenfunctions define a new family of orthogonal polynomials in two variables, $(r, \rho^2)$. These polynomials are related to the finite-dimensional representations of the algebra $gl(2) \ltimes {\it R}^3 \in g^{(2)}$ (discovered by S Lie around 1880 which went almost unnoticed), which occurs as the hidden algebra of the $G_2$ rational integrable system of 3 bodies on the line with 2- and 3-body interactions (the Wolfes model). Namely, those polynomials occur intrinsically in the study of the Zeeman effect on Hydrogen atom. It is shown that in the variables $(r, \rho, \varphi)$ in the quasi-exactly-solvable, generalized Coulomb problem new polynomial eigenfunctions in $(r, \rho^2)$-variables are found.