{
  "id": "2212.03108",
  "version": "v1",
  "published": "2022-12-02T20:11:17.000Z",
  "updated": "2022-12-02T20:11:17.000Z",
  "title": "Two-body Coulomb problem and $g^{(2)}$ algebra (once again about the Hydrogen atom)",
  "authors": [
    "Alexander V Turbiner",
    "Adrian M Escobar Ruiz"
  ],
  "comment": "7 pages, 3 figures",
  "categories": [
    "quant-ph",
    "math.AP",
    "math.RT",
    "physics.atom-ph"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-02T20:11:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hydrogen atom",
      "two-body coulomb problem",
      "polynomials occur",
      "three-dimensional system",
      "wolfes model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}