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

• URL: http://arxiv.org/abs/2212.03108v1
• Date: Fri, 2 Dec 2022 20:11:17 GMT
• Title: Two-body Coulomb problem and $g^{(2)}$ algebra (once again about the Hydrogen atom)
• Authors: Alexander V Turbiner and Adrian M Escobar Ruiz
• Abstract summary: It is shown that if the symmetry of the three-dimensional system is $(r, rho, varphi)$, the variables $(r, rho, varphi)$ allow a separation of variable $varphi$ and eigenfunctions. Thoses occur in the study of the Zeeman effect on Hydrogen atom.
• Score: 77.34726150561087
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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