Rank one perturbations in a Pontryagin space with one negative square
The interest in extending the study of rank one perturbations from Hilbert to Pontryagin spaces is justified by some recent results concerning the point interactions in quantum mechanical models. The framework of the present paper is a Pontryagin space \(({\mathfrak H},[.,.])\) of negative index one. The perturbations of a selfadjoint operator \(A\) on \({\mathfrak H}\) have the form \(A(t)=A+ t[.,\omega]\omega\), where \(w\in{\mathcal H}\) and \(t\in \mathbb{R}\). The spectral analysis of these perturbations is based on a subclass \(N_{0,1}\) of the generalized Nevanlinna functions, for which \(\lim_{y\to \infty} Q(iy)/y=0\) and \(\limsup_{y\to\infty} y|\text{Im} Q(iy)|<\infty\). Most results refer to the explicit characterization of the critical eigenvalues of \(A(t)\) and to the root subspaces of the Friedrichs extension \(S_F=S_+(\{0\} \oplus\text{span}\{\omega\})\), where \(S\) is the restriction of \(A\) to \(\text{dom} S =\{f\in\text{dom} A:[f,\omega]= 0\}\). The last section of the paper contains examples encountered in mathematical physics.
scientific article; zbMATH DE number 1763776
scientific article