{
  "id": "1505.07236",
  "version": "v1",
  "published": "2015-05-27T09:34:29.000Z",
  "updated": "2015-05-27T09:34:29.000Z",
  "title": "Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces",
  "authors": [
    "A. Mantile",
    "A. Posilicano",
    "M. Sini"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.FA",
    "math.MP"
  ],
  "abstract": "The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic operator on $\\mathbb{R}% ^{n}$ with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Krein-like resolvent formulas where the reference operator coincides with the \"free\" operator with domain $H^{2}(\\mathbb{R}^{n})$; this provides an useful tool for the scattering problem from a hypersurface. Concrete examples of this construction are developed in connection with the standard boundary conditions, Dirichlet, Neumann, Robin, $\\delta$ and $\\delta^{\\prime}$-type, assigned either on a $n-1$ dimensional compact boundary $\\Gamma=\\partial\\Omega$ or on a relatively open part $\\Sigma\\subset\\Gamma$. Schatten-von Neumann estimates for the difference of the powers of resolvents of the free and the perturbed operators are also proven; these give existence and completeness of the wave operators of the associated scattering systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-27T09:34:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "self-adjoint elliptic operators",
      "closed hypersurfaces",
      "relatively open part",
      "second-order elliptic operator",
      "construct self-adjoint realizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}