Self-adjoint Operators with Inner Singularities and Pontryagin Spaces

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 m where the direct summand space C m ⊕ C m is 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.