{
  "id": "1104.2585",
  "version": "v2",
  "published": "2011-04-13T19:40:05.000Z",
  "updated": "2011-05-23T18:16:41.000Z",
  "title": "Generalized Kepler Problems I: Without Magnetic Charges",
  "authors": [
    "Guowu Meng"
  ],
  "comment": "30 pages",
  "journal": "J. Math. Phys. 54, 012109(2013)",
  "doi": "10.1063/1.4775343",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "For each simple euclidean Jordan algebra $V$ of rank $\\rho$ and degree $\\delta$, we introduce a family of classical dynamic problems. These dynamical problems all share the characteristic features of the Kepler problem for planetary motions, such as existence of Laplace-Runge-Lenz vector and hidden symmetry. After suitable quantizations, a family of quantum dynamic problems, parametrized by the nontrivial Wallach parameter $\\nu$, is obtained. Here, $\\nu\\in{\\mathcal W}(V):=\\{k {\\delta\\over 2}\\mid k=1, ..., (\\rho-1)\\}\\cup((\\rho-1){\\delta\\over 2}, \\infty)$ and was introduced by N. Wallach to parametrize the set of nontrivial scalar-type unitary lowest weight representations of the conformal group of $V$. For the quantum dynamic problem labelled by $\\nu$, the bound state spectra is $-{1/2\\over (I+\\nu{\\rho\\over 2})^2}$, I=0, 1, ... and its Hilbert space of bound states gives a new realization for the afore-mentioned representation labelled by $\\nu$. A few results in the literature about these representations become more explicit and more refined. The Lagrangian for a classical Kepler-type dynamic problem introduced here is still of the simple form: ${1\\over 2} ||\\dot x||^2+{1\\over r}$. Here, $\\dot x$ is the velocity of a unit-mass particle moving on the space consisting of $V$'s semi-positive elements of a fixed rank, and $r$ is the inner product of $x$ with the identity element of $V$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-05-23T18:16:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generalized kepler problems",
      "magnetic charges",
      "quantum dynamic problem",
      "scalar-type unitary lowest weight representations",
      "nontrivial scalar-type unitary lowest weight"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AIP",
      "journal": "J. Math. Phys."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.2585M"
    }
  }
}