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Operator Theory and Related Topics pp 105–175Cite as

Birkhäuser

Self-adjoint Operators with Inner Singularities and Pontryagin Spaces

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 118))

Abstract

Let A 0 be an unbounded self-adjoint operator in a Hilbert space H 0 and let χ be a generalized element of order — m — 1 in the rigging associated with A 0 and the inner product 〈·, ·〉0 of H 0. In [S1, S2, S3] operators H t, t · R ∪ ∞, are defined which serve as an interpretation for the family of operators A 0 + t -1 〈·, χ〉0 χ. The second summand here contains the inner singularity mentioned in the title. The operators H t act in Pontryagin spaces of the form π m = H 0⊕C m⊕C mwhere the direct summand space Cm ⊕ Cmis provided with an indefinite inner product. They can be interpreted both as a canonical extension of some symmetric operator S in π m and also as extensions of a one-dimensional restriction S 0 of A 0 in H 0 and hence they can be characterized by a class of Straus extensions of S 0 as well as via M.G. Krein’s formulas for (generalized) resolvents. In this paper we describe both these realizations explicitly and study their spectral properties. A main role is played by a special class of Q-functions. Factorizations of these functions correspond to the separation of the nonpositive type spectrum from the positive spectrum of H t. As a consequence, in Subsection 7.3 a family of self-adjoint Hilbert space operators is obtained which can serve as a nontrivial quantum model associated with the operators A 0 + t -1 〈·, χ〉0 χ.

*

The research of Heinz Langer was supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 12176 MAT.

The research of Yuri Shondin was supported by the Netherlands Organization for Scientific Research NWO (NB 61-377) and by INTAS (93-0249-EXT).

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Groningen, P.O. Box 800, 9700, AV Groningen, The Netherlands

    Aad Dijksma

  2. Department of Mathematics, Technical University of Vienna, Wiedner Haupsstrasse 8-10/1411, A-1040, Vienna, Austria

    Heinz Langer

  3. Department of Theoretical Physics, State Pedagogical University, Str. Uly’anova 1, Nizhny Novgorod, 603600, Russia

    Yuri Shondin

  4. Department of Mathematics, Free University, De Boelelaan 1081a, 1081, HV Amsterdam, The Netherlands

    Chris Zeinstra

Authors
  1. Aad Dijksma
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  2. Heinz Langer
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  3. Yuri Shondin
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  4. Chris Zeinstra
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Editor information

Editors and Affiliations

  1. Department of Theoretical Physics, University of Odessa, 270026, Odessa, Ukraine

    V. M. Adamyan

  2. Department of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Ramat Aviv, Israel

    I. Gohberg

  3. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    M. Gorbachuk  & V. Gorbachuk  & 

  4. Department of Mathematics, Vrije Universiteit, De Boelelaan 1081a, 1081, HV Amsterdam, The Netherlands

    M. A. Kaashoek

  5. Department of Mathematics, Technical University of Vienna, Wiedner Hauptstrasse 8-10/1411, 1040, Vienna, Austria

    H. Langer

  6. Institute of Mathematics, Economics and Mechanics, Odessa State University, 270057, Dvoryanskaya str. 2, Odessa, Ukraine

    G. Popov

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© 2000 Springer Basel AG

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Dijksma, A., Langer, H., Shondin, Y., Zeinstra, C. (2000). Self-adjoint Operators with Inner Singularities and Pontryagin Spaces. In: Adamyan, V.M., et al. Operator Theory and Related Topics. Operator Theory: Advances and Applications, vol 118. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8413-6_8

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  • DOI: https://doi.org/10.1007/978-3-0348-8413-6_8

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9557-6

  • Online ISBN: 978-3-0348-8413-6

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