{
  "id": "2006.14757",
  "version": "v1",
  "published": "2020-06-26T02:21:15.000Z",
  "updated": "2020-06-26T02:21:15.000Z",
  "title": "Painlevé V and the Hankel Determinant for a Singularly Perturbed Jacobi Weight",
  "authors": [
    "Chao Min",
    "Yang Chen"
  ],
  "comment": "28 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the Hankel determinant generated by a singularly perturbed Jacobi weight $$ w(x,t):=(1-x^2)^\\alpha\\mathrm{e}^{-\\frac{t}{x^{2}}},\\;\\;\\;\\;\\;\\;x\\in[-1,1],\\;\\;\\alpha>0,\\;\\;t\\geq 0. $$ If $t=0$, it is reduced to the classical symmetric Jacobi weight. For $t>0$, the factor $\\mathrm{e}^{-\\frac{t}{x^{2}}}$ induces an infinitely strong zero at the origin. This Hankel determinant is related to the Wigner time-delay distribution in chaotic cavities. In the finite $n$ dimensional case, we obtain two auxiliary quantities $R_n(t)$ and $r_n(t)$ by using the ladder operator approach. We show that the Hankel determinant has an integral representation in terms of $R_n(t)$, where $R_n(t)$ is closely related to a particular Painlev\\'{e} V transcendent. Furthermore, we derive a second-order nonlinear differential equation and also a second-order difference equation for the logarithmic derivative of the Hankel determinant. This quantity can be expressed in terms of the Jimbo-Miwa-Okamoto $\\sigma$-function of a particular Painlev\\'{e} V. Then we consider the asymptotics of the Hankel determinant under a suitable double scaling, i.e. $n\\rightarrow\\infty$ and $t\\rightarrow 0$ such that $s=2n^2 t$ is fixed. Based on previous results by using the Coulomb fluid method, we obtain the large $s$ and small $s$ asymptotic behaviors of the scaled Hankel determinant, including the constant term in the asymptotic expansion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-26T02:21:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "41A60",
      "42C05"
    ],
    "keywords": [
      "hankel determinant",
      "singularly perturbed jacobi weight",
      "second-order nonlinear differential equation",
      "asymptotic",
      "coulomb fluid method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}