{
  "id": "2007.03649",
  "version": "v1",
  "published": "2020-07-07T17:37:53.000Z",
  "updated": "2020-07-07T17:37:53.000Z",
  "title": "The essential numerical range and a theorem of Simon on the absorption of eigenvalues",
  "authors": [
    "Brian Lins"
  ],
  "categories": [
    "math.FA",
    "math.SP"
  ],
  "abstract": "Let $A(t)$ be a holomorphic family of self-adjoint operators of type (B) on a complex Hilbert space $\\mathcal{H}$. Kato-Rellich perturbation theory says that isolated eigenvalues of $A(t)$ will be analytic functions of $t$ as long as they remain below the minimum of the essential spectrum of $A(t)$. At a threshold value $t_0$ where one of these eigenvalue functions hits the essential spectrum, the corresponding point in the essential spectrum might or might not be an eigenvalue of $A(t_0)$. Our results generalize a theorem of Simon to give a sufficient condition for the minimum of the essential spectrum to be an eigenvalue of $A(t_0)$ based on the rate at which eigenvalues approach the essential spectrum. We also show that the rates at which the eigenvalues of $A(t)$ can approach the essential spectrum from below correspond to eigenvalues of a bounded self-adjoint operator. The key insight behind these results is the essential numerical range which was recently extended to unbounded operators by B\\\"{o}gli, Marletta, and Tretter.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-07T17:37:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A55",
      "47A07",
      "47A12"
    ],
    "keywords": [
      "essential numerical range",
      "essential spectrum",
      "absorption",
      "kato-rellich perturbation theory says",
      "eigenvalue functions hits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}