On the perturbation theory of self-adjoint operators
Let \(A\) be a semi-bounded selfadjoint operator on a separable Hilbert space \(\mathcal{H}\). A selfadjoint operator \(\tilde{A} \neq A\) on \(\mathcal{H}\) is called regular perturbation of \(A\) if \(D(A) \cap D(\tilde{A}) =D(A)\) and singular perturbation otherwise, where \(D(A)\) denotes the domain of the operator. Furthermore, a singular perturbation \(\tilde{A}\) is said to be purely singular if the maximal common part of \(A\) and \(\tilde{A}\) is densely defined, and mixed singular otherwise. Compared to mixed singular perturbations, regular and purely singular perturbations are well-established by the additive perturbation theory and the extension theory, respectively. In this paper, the authors show that all types of perturbations (regular, purely singular, and mixed singular) can be described and studied in both the frameworks of extension theory and the additive perturbation theory. Their method is to develop a bijection between selfadjoint operators \(\tilde{A}\) and the set of pairs \(\{N, T\}\), where \(N\) is a subspace of \(\mathcal{H}\) and \(T\) is an admissible selfadjoint operator in \(N\), and then to establish connection between \(\{N, T\}\) and the parameters in the frameworks of the extension theory and the additive perturbation theory. They also use the correspondence \(\tilde{A} \leftrightarrow \{N, T\}\) to show that any singular finite rank perturbation can be approximated by regular finite rank perturbations in the norm resolvent sense. Lastly, an application is given to the study of Schrödinger operators with point interactions.
scientific article; zbMATH DE number 5545412
scientific article