{
  "id": "math/0102018",
  "version": "v2",
  "published": "2001-02-02T16:45:05.000Z",
  "updated": "2002-05-23T09:14:59.000Z",
  "title": "Boundary Conditions for Singular Perturbations of Self-Adjoint Operators",
  "authors": [
    "Andrea Posilicano"
  ],
  "comment": "Revised version. To appear in Operator Theory: Advances and Applications, vol. 132",
  "categories": [
    "math.FA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $A:D(A)\\subseteq\\H\\to\\H$ be an injective self-adjoint operator and let $\\tau:D(A)\\to\\X$, X a Banach space, be a surjective linear map such that $\\|\\tau\\phi\\|_\\X\\le c \\|A\\phi\\|_\\H$. Supposing that \\text{\\rm Range}$ (\\tau')\\cap\\H' =\\{0\\}$, we define a family $A^\\tau_\\Theta$ of self-adjoint operators which are extensions of the symmetric operator $A_{|\\{\\tau=0\\}.}$. Any $\\phi$ in the operator domain $D(A^\\tau_\\Theta)$ is characterized by a sort of boundary conditions on its univocally defined regular component $\\phireg$, which belongs to the completion of D(A) w.r.t. the norm $\\|A\\phi\\|_\\H$. These boundary conditions are written in terms of the map $\\tau$, playing the role of a trace (restriction) operator, as $\\tau\\phireg=\\Theta Q_\\phi$, the extension parameter $\\Theta$ being a self-adjoint operator from X' to X. The self-adjoint extension is then simply defined by $A^\\tau_\\Theta\\phi:=A \\phireg$. The case in which $A\\phi=T*\\phi$ is a convolution operator on LD, T a distribution with compact support, is studied in detail.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2002-05-23T09:14:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boundary conditions",
      "singular perturbations",
      "compact support",
      "injective self-adjoint operator",
      "symmetric operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......2018P"
    }
  }
}