{
  "id": "0903.5277",
  "version": "v1",
  "published": "2009-03-30T18:16:58.000Z",
  "updated": "2009-03-30T18:16:58.000Z",
  "title": "Self-adjoint extensions and spectral analysis in Calogero problem",
  "authors": [
    "D. M. Gitman",
    "I. V. Tyutin",
    "B. L. Voronov"
  ],
  "comment": "39 pages",
  "categories": [
    "quant-ph",
    "hep-th",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper, we present a mathematically rigorous quantum-mechanical treatment of a one-dimensional motion of a particle in the Calogero potential $\\alpha x^{-2}$. Although the problem is quite old and well-studied, we believe that our consideration, based on a uniform approach to constructing a correct quantum-mechanical description for systems with singular potentials and/or boundaries, proposed in our previous works, adds some new points to its solution. To demonstrate that a consideration of the Calogero problem requires mathematical accuracy, we discuss some \"paradoxes\" inherent in the \"naive\" quantum-mechanical treatment. We study all possible self-adjoint operators (self-adjoint Hamiltonians) associated with a formal differential expression for the Calogero Hamiltonian. In addition, we discuss a spontaneous scale-symmetry breaking associated with self-adjoint extensions. A complete spectral analysis of all self-adjoint Hamiltonians is presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-30T18:16:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "calogero problem",
      "self-adjoint extensions",
      "self-adjoint hamiltonians",
      "formal differential expression",
      "complete spectral analysis"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1088/1751-8113/43/14/145205",
      "journal": "Journal of Physics A Mathematical General",
      "year": 2010,
      "month": "Apr",
      "volume": 43,
      "number": 14,
      "pages": 145205
    },
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010JPhA...43n5205G",
      "inspire": 1370022
    }
  }
}