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

Alexander V Turbiner, Adrian M Escobar Ruiz

Published 2022-12-02Version 1

Taking the Hydrogen atom as an example it is shown that if the symmetry of the three-dimensional system is $O(2) \oplus Z_2$, the variables $(r, \rho, \varphi)$ allow a separation of variable $\varphi$ and the eigenfunctions define a new family of orthogonal polynomials in two variables, $(r, \rho^2)$. These polynomials are related with the finite-dimensional representations of the algebra $gl(2) \ltimes {\it R}^3 \in g^{(2)}$, which occurs as the hidden algebra of the $G_2$ rational integrable system of 3 bodies on the line (the Wolfes model). Namely, those polynomials occur in the study of the Zeeman effect on Hydrogen atom.