On supersingular perturbations of non-semibounded self-adjoint operators

In this paper self-adjoint realizations of the formal expression Aα:=A+α⟨ϕ,⋅⟩ϕ are described, where α∈R∪{∞}, the operator A is self-adjoint in a Hilbert space H and ϕ is a supersingular element from the scale space H−n−2(A)∖H−n−1(A) for n⩾1. The crucial point is that the spectrum of A may consist of the whole real line. We construct two models to describe the family (Aα). It can be interpreted in a Hilbert space with a twisted version of Krein's formula, or with a more classical version of Krein's formula but in a Pontryagin space. Finally, we compare the two approaches in terms of the respective Q-functions.