Scattering problem in the theory of singular perturbations of self-adjoint operators
Let \(H_ 0>0\) be a selfadjoint operator and let H be a perturbed operator of \(H_ 0\) with singular perturbation. Suppose that \(H_ 0\) and H act in different Hilbert spaces \({\mathfrak H}\) and \({\mathfrak H}'\), where the identification operator J:\({\mathfrak H}\to {\mathfrak H}'\), defined on a dense set \(\Phi\), is not necessarily bounded. This paper is devoted to the study of the corresponding wave operator \(W_{\pm}(H_ 0,H,J,D)=W_{\pm},\) where D is a fixed subset of \(\Phi\). It contains many results without proofs about the existence and closedness of \(W_{\pm}\), the existence of the wave operator \(W_{\pm}(\phi (H_ 0),\phi (H),J,D)=W_{\pm \phi}\) for an admissible function \(\phi\), and about the question when \(W_{\pm \phi}H_{\pm}\) holds. Finally, the applicability of the nuclear methods and the method of orthogonal extension are considered.
scientific article; zbMATH DE number 3884802
scientific article