{
  "id": "1901.09289",
  "version": "v1",
  "published": "2019-01-26T22:48:55.000Z",
  "updated": "2019-01-26T22:48:55.000Z",
  "title": "Inverse Scattering for the Laplace operator with boundary conditions on Lipschitz surfaces",
  "authors": [
    "Andrea Mantile",
    "Andrea Posilicano"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We provide a general scheme, in the combined frameworks of Mathematical Scattering Theory and Factorization Method, for inverse scattering for the couple of self-adjoint operators $(\\widetilde\\Delta,\\Delta)$, where $\\Delta$ is the free Laplacian in $L^{2}({\\mathbb R}^{3})$ and $\\widetilde\\Delta$ is one of its singular perturbations, i.e., such that the set $\\{u\\in H^{2}({\\mathbb R}^{3})\\cap \\text{dom}(\\widetilde\\Delta)\\, :\\, \\Delta u=\\widetilde\\Delta u\\}$ is dense. Typically $\\widetilde\\Delta$ corresponds to a self-adjoint realization of the Laplace operator with some kind of boundary conditions imposed on a null subset; in particular our results apply to standard, either separating or semi-transparent, boundary conditions at $\\Gamma=\\partial\\Omega$, where $\\Omega\\subset{\\mathbb R}^{3}$ is a bounded Lipschitz domain. Similar results hold in the case the boundary conditions are assigned only on $\\Sigma\\subset\\Gamma$, a relatively open subset with a Lipschitz boundary. We show that either $\\Gamma$ or $\\Sigma$ are determined by the knowledge of the Scattering Matrix, equivalently of the Far Field Operator, at a single frequency.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-26T22:48:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boundary conditions",
      "laplace operator",
      "inverse scattering",
      "lipschitz surfaces",
      "far field operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}