Ontology type: schema:Chapter
2000
AUTHORSAad Dijksma , Heinz Langer , Yuri Shondin , Chris Zeinstra
ABSTRACTLet 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... »
PAGES105-175
Operator Theory and Related Topics
ISBN
978-3-0348-9557-6
978-3-0348-8413-6
http://scigraph.springernature.com/pub.10.1007/978-3-0348-8413-6_8
DOIhttp://dx.doi.org/10.1007/978-3-0348-8413-6_8
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