{
  "id": "quant-ph/0203121",
  "version": "v1",
  "published": "2002-03-25T08:30:24.000Z",
  "updated": "2002-03-25T08:30:24.000Z",
  "title": "Quantum models related to fouled Hamiltonians of the harmonic oscillator",
  "authors": [
    "P. Tempesta",
    "E. Alfinito",
    "R. A. Leo",
    "G. Soliani"
  ],
  "comment": "24 pages, no figures, accepted for publication on JMP",
  "journal": "J.Math.Phys. 43 (2002) 3538-3553",
  "categories": [
    "quant-ph",
    "hep-th"
  ],
  "abstract": "We study a pair of canonoid (fouled) Hamiltonians of the harmonic oscillator which provide, at the classical level, the same equation of motion as the conventional Hamiltonian. These Hamiltonians, say $K_{1}$ and $K_{2}$, result to be explicitly time-dependent and can be expressed as a formal rotation of two cubic polynomial functions, $H_{1}$ and $H_{2}$, of the canonical variables (q,p). We investigate the role of these fouled Hamiltonians at the quantum level. Adopting a canonical quantization procedure, we construct some quantum models and analyze the related eigenvalue equations. One of these models is described by a Hamiltonian admitting infinite self-adjoint extensions, each of them has a discrete spectrum on the real line. A self-adjoint extension is fixed by choosing the spectral parameter $\\epsilon$ of the associated eigenvalue equation equal to zero. The spectral problem is discussed in the context of three different representations. For $\\epsilon =0$, the eigenvalue equation is exactly solved in all these representations, in which square-integrable solutions are explicity found. A set of constants of motion corresponding to these quantum models is also obtained. Furthermore, the algebraic structure underlying the quantum models is explored. This turns out to be a nonlinear (quadratic) algebra, which could be applied for the determination of approximate solutions to the eigenvalue equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-03-25T08:30:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03.65.Ge",
      "03.65.Fd"
    ],
    "keywords": [
      "harmonic oscillator",
      "quantum models",
      "fouled hamiltonians",
      "hamiltonian admitting infinite self-adjoint extensions",
      "eigenvalue equation equal"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AIP",
      "journal": "Journal of Mathematical Physics",
      "doi": "10.1063/1.1479300",
      "year": 2002,
      "month": "Jul",
      "volume": 43,
      "number": 7,
      "pages": 3538
    },
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 584681,
      "adsabs": "2002JMP....43.3538T"
    }
  }
}