{
  "id": "math-ph/0402022",
  "version": "v1",
  "published": "2004-02-10T08:01:31.000Z",
  "updated": "2004-02-10T08:01:31.000Z",
  "title": "Lower limit in semiclassical form for the number of bound states in a central potential",
  "authors": [
    "Fabian Brau",
    "Francesco Calogero"
  ],
  "comment": "9 pages",
  "journal": "Phys. Lett. A321, 225 (2004)",
  "doi": "10.1016/j.physleta.2003.12.034",
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "We identify a class of potentials for which the semiclassical estimate $N^{\\text{(semi)}}=\\frac{1}{\\pi}\\int_0^\\infty dr\\sqrt{-V(r)\\theta[-V(r)]}$ of the number $N$ of (S-wave) bound states provides a (rigorous) lower limit: $N\\ge {{N^{\\text{(semi)}}}}$, where the double braces denote the integer part. Higher partial waves can be included via the standard replacement of the potential $V(r)$ with the effective $\\ell$-wave potential $V_\\ell^{\\text{(eff)}}(r)=V(r)+\\frac{\\ell(\\ell+1)}{r^2}$. An analogous upper limit is also provided for a different class of potentials, which is however quite severely restricted.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-02-10T08:01:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bound states",
      "lower limit",
      "central potential",
      "semiclassical form",
      "higher partial waves"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}