Self-adjoint operators with inner singularities and pontryagin spaces

Let A(0) be an unbounded self-adjoint operator in a Hilbert space H-0 and let chi be a generalized element of order -m -1 in the rigging associated with Ag and the inner product (., .)(0) of H-0. In [S1, S2, S3] operators H-t, t epsilon R U {infinity}, are defined which serve as an interpretation for the family of operators A(0) + t(-1)(. , chi)(0) chi. The second summand here contains the inner singularity mentioned in the title. The operators H-t act in Pontryagin spaces of the form Pi(m) = H(0)circle plus C-m circle plus C-m where the direct summand space C-m circle plus C-m is provided with an indefinite inner product. They can be interpreted both as a canonical extension of some symmetric operator S in Pi(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 Ag + t(-1)(. , chi)(0) chi.