Self-adjoint Operators with Inner Singularities and Pontryagin Spaces View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

2000

AUTHORS

Aad Dijksma , Heinz Langer , Yuri Shondin , Chris Zeinstra

ABSTRACT

Let A0 be an unbounded self-adjoint operator in a Hilbert space H0 and let χ be a generalized element of order — m — 1 in the rigging associated with A0 and the inner product 〈·, ·〉0 of H0. In [S1, S2, S3] operators Ht, t · R ∪ ∞, are defined which serve as an interpretation for the family of operators A0 + t-1 〈·, χ〉0 χ. The second summand here contains the inner singularity mentioned in the title. The operators Ht act in Pontryagin spaces of the form πm = H0⊕Cm⊕Cmwhere 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 πm and also as extensions of a one-dimensional restriction S0 of A0 in H0 and hence they can be characterized by a class of Straus extensions of S0 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 Ht. 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 A0 + t-1 〈·, χ〉0 χ. More... »

PAGES

105-175

Book

TITLE

Operator Theory and Related Topics

ISBN

978-3-0348-9557-6
978-3-0348-8413-6

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-0348-8413-6_8

DOI

http://dx.doi.org/10.1007/978-3-0348-8413-6_8

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1000870627


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