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

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.