{
  "id": "1507.02123",
  "version": "v1",
  "published": "2015-07-08T12:27:26.000Z",
  "updated": "2015-07-08T12:27:26.000Z",
  "title": "Strong coupling asymptotics for Schrödinger operators with an interaction supported by an open arc in three dimensions",
  "authors": [
    "Pavel Exner",
    "Sylwia Kondej"
  ],
  "comment": "17 pages, no figures",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP",
    "quant-ph"
  ],
  "abstract": "We consider Schr\\\"odinger operators with a strongly attractive singular interaction supported by a finite curve $\\Gamma$ of lenghth $L$ in $\\R^3$. We show that if $\\Gamma$ is $C^4$-smooth and has regular endpoints, the $j$-th eigenvalue of such an operator has the asymptotic expansion $\\lambda_j (H_{\\alpha,\\Gamma})= \\xi_\\alpha +\\lambda _j(S)+\\mathcal{O}(\\mathrm{e}^{\\pi \\alpha })$ as the coupling parameter $\\alpha\\to\\infty$, where $\\xi_\\alpha = -4\\,\\mathrm{e}^{2(-2\\pi\\alpha +\\psi(1))}$ and $\\lambda _j(S)$ is the $j$-th eigenvalue of the Schr\\\"odinger operator $S=-\\frac{\\D^2}{\\D s^2 }- \\frac14 \\gamma^2(s)$ on $L^2(0,L)$ with Dirichlet condition at the interval endpoints in which $\\gamma$ is the curvature of $\\Gamma$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-08T12:27:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q10",
      "35J10",
      "35P20"
    ],
    "keywords": [
      "strong coupling asymptotics",
      "schrödinger operators",
      "open arc",
      "th eigenvalue",
      "dimensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150702123E"
    }
  }
}