{
  "id": "math/0005082",
  "version": "v3",
  "published": "2000-05-09T09:46:00.000Z",
  "updated": "2000-12-14T16:08:20.000Z",
  "title": "A Krein-Like Formula for Singular Perturbations of Self-Adjoint Operators and Applications",
  "authors": [
    "Andrea Posilicano"
  ],
  "comment": "Proposition 2.1 revised. Remarks 2.15 and 2.16 added. 38 pages. To appear in Journal of Functional Analysis",
  "categories": [
    "math.FA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Given a self-adjoint operator $A:D(A)\\subseteq\\calH\\to\\calH$ and a continuous linear operator $\\tau:D(A)\\to\\X$ with Range$ \\tau'\\cap\\calH' ={0}$, $\\X$ a Banach space, we explicitly construct a family $A^\\tau_\\Theta$ of self-adjoint operators such that any $A^\\tau_\\Theta$ coincides with the original $A$ on the kernel of $\\tau$. Such a family is obtained by giving a Kre\\u\\i n-like formula where the role of the deficiency spaces is played by the dual pair $(\\X,\\X')$; the parameter $\\Theta$ belongs to the space of symmetric operators from $\\X'$ to $\\X$. When $\\X=\\C$ one recovers the ``$\\calH_{-2}$ -construction'' of Kiselev and Simon and so, to some extent, our results can be regarded as an extension of it to the infinite rank case. Considering the situation in which $\\calH=L^2(\\RE^n)$ and $\\tau$ is the trace (restriction) operator along some null subset, we give various applications to singular perturbations of non necessarily elliptic pseudo-differential operators, thus unifying and extending previously known results.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2000-12-14T16:08:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "self-adjoint operator",
      "singular perturbations",
      "krein-like formula",
      "applications",
      "non necessarily elliptic pseudo-differential operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......5082P"
    }
  }
}