On rank one \(H_{-3}\)-perturbations of positive self-adjoint operators (Q2702406)

Let \(A\) be a self-adjoint and positive operator in some Hilbert space \(H\) and denote by \(H_s(A)\), \(s\in\mathbb{R}\), the scale of Hilbert spaces associated with \(A\). For fixed \(\phi\in H_{-3}(A)\setminus H_{-2}(A)\) the authors analyze rank one perturbations \(A_\alpha\) of \(A\) formally given by \(A_\alpha= A+ \alpha\langle\phi,\cdot\rangle \phi\). To this aim they look at the restriction \(A^0\) of \(A\) to the space \(\{u\in H_3(A): \langle \phi,u\rangle= 0\}\). Since \(\phi\not\in H_{-2}(A)\), \(A^0\) is essentially self-adjoint which means that one can not use the formal definition of \(A_\alpha\) directly to give a reasonable definition of \(A_\alpha\). Therefore the one-dimensional extension \({\mathcal H}:= D(A^0)\dot+\mathbb{C}\) of \(D(A)\) is equipped with a scalar product and it turns out that the completion of \({\mathcal H}\) with respect to this scalar product coincides with \(H_1(A)\dot+ \mathbb{C}\). Having done these preparations, a family \(({\mathcal A}_\theta)_{\theta\in [0,2\pi)}\) in the completion of \({\mathcal H}\) is constructed and it is shown that each \({\mathcal A}_\theta\) is self-adjoint.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00049].