# Journal of Operator Theory

Volume 81, Issue 1, Winter 2019  pp. 195-223.

On supersingular perturbations of non-semibounded self-adjoint operators

Authors:  Pavel Kurasov (1), Annemarie Luger (2), Christoph Neuner (3)
Author institution: (1) Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden
(2) Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden
(3) Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden

Summary:  In this paper self-adjoint realizations of the formal expression $A_{\alpha} := A + \alpha\langle \phi, \cdot \rangle \phi$ are described, where $\alpha \in \mathbb{R} \cup \{\infty\}$, the operator $A$ is self-adjoint in a Hilbert space $\mathcal{H}$ and $\phi$ is a supersingular element from the scale space $\mathcal{H}_{-n - 2}(A) \backslash \mathcal{H}_{- n - 1}(A)$ for $n \geqslant 1$. The crucial point is that the spectrum of $A$ may consist of the whole real line. We construct two models to describe the family $(A_{\alpha})$. 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.

DOI: http://dx.doi.org/10.7900/jot.2017dec22.2183
Keywords:  unbounded self-adjoint operator, supersingular perturbation, generalized Nevanlinna function