{
  "id": "1009.0873",
  "version": "v1",
  "published": "2010-09-04T21:20:28.000Z",
  "updated": "2010-09-04T21:20:28.000Z",
  "title": "On a class of $J$-self-adjoint operators with empty resolvent set",
  "authors": [
    "Sergii Kuzhel",
    "Carsten Trunk"
  ],
  "journal": "J. Math. Anal. Appl. 379 (2011), no. 1, 272-289",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP",
    "quant-ph"
  ],
  "abstract": "In the present paper we investigate the set $\\Sigma_J$ of all $J$-self-adjoint extensions of a symmetric operator $S$ with deficiency indices $<2,2>$ which commutes with a non-trivial fundamental symmetry $J$ of a Krein space $(\\mathfrak{H}, [\\cdot,\\cdot])$, SJ=JS. Our aim is to describe different types of $J$-self-adjoint extensions of $S$. One of our main results is the equivalence between the presence of $J$-self-adjoint extensions of $S$ with empty resolvent set and the commutation of $S$ with a Clifford algebra ${\\mathcal C}l_2(J,R)$, where $R$ is an additional fundamental symmetry with $JR=-RJ$. This enables one to construct the collection of operators $C_{\\chi,\\omega}$ realizing the property of stable $C$-symmetry for extensions $A\\in\\Sigma_J$ directly in terms of ${\\mathcal C}l_2(J,R)$ and to parameterize the corresponding subset of extensions with stable $C$-symmetry. Such a situation occurs naturally in many applications, here we discuss the case of an indefinite Sturm-Liouville operator on the real line and a one dimensional Dirac operator with point interaction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-09-04T21:20:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A55",
      "47A57",
      "81Q15"
    ],
    "keywords": [
      "empty resolvent set",
      "self-adjoint operators",
      "self-adjoint extensions",
      "additional fundamental symmetry",
      "non-trivial fundamental symmetry"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.0873K"
    }
  }
}