{
  "id": "math-ph/0508032",
  "version": "v2",
  "published": "2005-08-15T14:47:29.000Z",
  "updated": "2005-11-25T16:37:35.000Z",
  "title": "Spectra of observables in the q-oscillator and q-analogue of the Fourier transform",
  "authors": [
    "Anatoliy Klimyk"
  ],
  "comment": "Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/",
  "journal": "SIGMA 1 (2005), 008, 17 pages",
  "doi": "10.3842/SIGMA.2005.008",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "Spectra of the position and momentum operators of the Biedenharn-Macfarlane q-oscillator (with the main relation aa^+-qa^+a=1) are studied when q>1. These operators are symmetric but not self-adjoint. They have a one-parameter family of self-adjoint extensions. These extensions are derived explicitly. Their spectra and eigenfunctions are given. Spectra of different extensions do not intersect. The results show that the creation and annihilation operators a^+ and a of the q-oscillator for q>1 cannot determine a physical system without further more precise definition. In order to determine a physical system we have to choose appropriate self-adjoint extensions of the position and momentum operators.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-11-25T16:37:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q10",
      "81S05",
      "41B15"
    ],
    "keywords": [
      "fourier transform",
      "q-analogue",
      "choose appropriate self-adjoint extensions",
      "observables",
      "momentum operators"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}