{
  "id": "0907.0650",
  "version": "v1",
  "published": "2009-07-03T14:54:44.000Z",
  "updated": "2009-07-03T14:54:44.000Z",
  "title": "On the unitary equivalence of absolutely continuous parts of self-adjoint extensions",
  "authors": [
    "Mark M. Malamud",
    "Hagen Neidhardt"
  ],
  "categories": [
    "math-ph",
    "math.FA",
    "math.MP"
  ],
  "abstract": "The classical Weyl-von Neumann theorem states that for any self-adjoint operator $A$ in a separable Hilbert space $\\mathfrak H$ there exists a (non-unique) Hilbert-Schmidt operator $C = C^*$ such that the perturbed operator $A+C$ has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering self-adjoint extensions of a given densely defined symmetric operator $A$ in $\\mathfrak H$ and fixing an extension $A_0 = A_0^*$. We show that for a wide class of symmetric operators the absolutely continuous parts of extensions $\\widetilde A = {\\widetilde A}^*$ and $A_0$ are unitarily equivalent provided that their resolvent difference is a compact operator. Namely, we show that this is true whenever the Weyl function $M(\\cdot)$ of a pair $\\{A,A_0\\}$ admits bounded limits $M(t) := \\wlim_{y\\to+0}M(t+iy)$ for a.e. $t \\in \\mathbb{R}$. This result is applied to direct sums of symmetric operators and Sturm-Liouville operators with operator potentials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-07-03T14:54:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A57",
      "47B25",
      "47A55"
    ],
    "keywords": [
      "absolutely continuous parts",
      "self-adjoint extensions",
      "unitary equivalence",
      "symmetric operator",
      "classical weyl-von neumann theorem states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.0650M"
    }
  }
}