# 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

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