{
  "id": "1508.04577",
  "version": "v1",
  "published": "2015-08-19T09:24:49.000Z",
  "updated": "2015-08-19T09:24:49.000Z",
  "title": "On absence of bound states for weakly attractive $δ^\\prime$-interactions supported on non-closed curves in $\\mathbb{R}^2$",
  "authors": [
    "Michal Jex",
    "Vladimir Lotoreichik"
  ],
  "comment": "23 pages, 2 figures",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "Let $\\Lambda\\subset\\mathbb{R}^2$ be a non-closed piecewise-$C^1$ curve, which is either bounded with two free endpoints or unbounded with one free endpoint. Let $u_\\pm|_\\Lambda \\in L^2(\\Lambda)$ be the traces of a function $u$ in the Sobolev space $H^1({\\mathbb R}^2\\setminus \\Lambda)$ onto two faces of $\\Lambda$. We prove that for a wide class of shapes of $\\Lambda$ the Schr\\\"odinger operator $\\mathsf{H}_\\omega^\\Lambda$ with $\\delta^\\prime$-interaction supported on $\\Lambda$ of strength $\\omega \\in L^\\infty(\\Lambda;\\mathbb{R})$ associated with the quadratic form \\[ H^1(\\mathbb{R}^2\\setminus\\Lambda)\\ni u \\mapsto \\int_{\\mathbb{R}^2}\\big|\\nabla u \\big|^2 \\mathsf{d} x - \\int_\\Lambda \\omega \\big| u_+|_\\Lambda - u_-|_\\Lambda \\big|^2 \\mathsf{d} s \\] has no negative spectrum provided that $\\omega$ is pointwise majorized by a strictly positive function explicitly expressed in terms of $\\Lambda$. If, additionally, the domain $\\mathbb{R}^2\\setminus\\Lambda$ is quasi-conical, we show that $\\sigma(\\mathsf{H}_\\omega^\\Lambda) = [0,+\\infty)$. For a bounded curve $\\Lambda$ in our class and non-varying interaction strength $\\omega\\in\\mathbb{R}$ we derive existence of a constant $\\omega_* > 0$ such that $\\sigma(\\mathsf{H}_\\omega^\\Lambda) = [0,+\\infty)$ for all $\\omega \\in (-\\infty, \\omega_*]$; informally speaking, bound states are absent in the weak coupling regime.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-19T09:24:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bound states",
      "non-closed curves",
      "weakly attractive",
      "free endpoint",
      "positive function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}