{
  "id": "2409.03900",
  "version": "v1",
  "published": "2024-09-05T20:19:25.000Z",
  "updated": "2024-09-05T20:19:25.000Z",
  "title": "Demkov--Fradkin tensor for curved harmonic oscillators",
  "authors": [
    "Şengül Kuru",
    "Javier Negro",
    "Sergio Salamanca"
  ],
  "comment": "25 pages, 10 figures",
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "In this work, we obtain the Demkov-Fradkin tensor of symmetries for the quantum curved harmonic oscillator in a space with constant curvature given by a parameter $\\kappa$. In order to construct this tensor we have firstly found a set of basic operators which satisfy the following conditions: i) their products give symmetries of the problem; in fact the Hamiltonian is a combination of such products; ii) they generate the space of eigenfunctions as well as the eigenvalues in an algebraic way; iii) in the limit of zero curvature, they come into the well known creation/annihilation operators of the flat oscillator. The appropriate products of such basic operators will produce the curved Demkov-Fradkin tensor. However, these basic operators do not satisfy Heisenberg commutators but close another Lie algebra. As a by-product, the classical Demkov-Fradkin tensor for the classical curved harmonic oscillator has been obtained by the same method. The case of two dimensions has been worked out in detail: the operators close a $so_\\kappa(4)$ Lie algebra; the spectrum and eigenfunctions are explicitly solved in an algebraic way and in the classical case the trajectories have been computed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-05T20:19:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "basic operators",
      "algebraic way",
      "lie algebra",
      "quantum curved harmonic oscillator",
      "satisfy heisenberg commutators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}