{
  "id": "1304.7696",
  "version": "v1",
  "published": "2013-04-29T16:00:34.000Z",
  "updated": "2013-04-29T16:00:34.000Z",
  "title": "Spectral asymptotics of a strong $δ'$ interaction on a planar loop",
  "authors": [
    "Pavel Exner",
    "Michal Jex"
  ],
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP",
    "quant-ph"
  ],
  "abstract": "We consider a generalized Schr\\\"odinger operator in $L^2(\\R^2)$ with an attractive strongly singular interaction of $\\delta'$ type characterized by the coupling parameter $\\beta>0$ and supported by a $C^4$-smooth closed curve $\\Gamma$ of length $L$ without self-intersections. It is shown that in the strong coupling limit, $\\beta\\to 0_+$, the number of eigenvalues behaves as $\\frac{2L}{\\pi\\beta} + \\OO(|\\ln\\beta|)$, and furthermore, that the asymptotic behaviour of the $j$-th eigenvalue in the same limit is $-\\frac{4}{\\beta^2} +\\mu_j+\\OO(\\beta|\\ln\\beta|)$, where $\\mu_j$ is the $j$-th eigenvalue of the Schr\\\"odinger operator on $L^2(0,L)$ with periodic boundary conditions and the potential $-\\frac14 \\gamma^2$ where $\\gamma$ is the signed curvature of $\\Gamma$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-29T16:00:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q10",
      "35J10",
      "35P15"
    ],
    "keywords": [
      "spectral asymptotics",
      "planar loop",
      "th eigenvalue",
      "periodic boundary conditions",
      "asymptotic behaviour"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1088/1751-8113/46/34/345201",
      "journal": "Journal of Physics A Mathematical General",
      "year": 2013,
      "month": "Aug",
      "volume": 46,
      "number": 34,
      "pages": 345201
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013JPhA...46H5201E"
    }
  }
}