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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