Pseudo-Hermiticity and theory of singular perturbations
A linear densely defined operator \(\widetilde{A}\) acting in a Hilbert space \(H\) is said to be pseudo-Hermitian if there exists a bounded Hermitian operator \(\eta\) with the properties \(0\in \rho(\eta)\) and \(\widetilde{A}^*= \eta \widetilde{A}\eta^{-1}\). Let \(A\) be a positive Hermitian operator and \(\eta\) be an invertible bounded Hermitian in \(H\) such that \(A\eta=\eta A.\) It is considered the formal expression \(A_V=A+V\), \(V:H_2\to H_{-2}\), \(H_{2}=\{u\in D(A)\mid \|(A+I)u\|<\infty\}\), where \(H_{-2}\) is the completion of \(H\) with respect to the norm \(\|f\|_{-2}=\|(A+I)^{-1}f\|\) under the asumption \(\overline{R(V)}\cap H=\{0\}.\) The operator \(A_0:=A_V|_{\ker V}=A|_{\ker V}\), \(D(A_0)=\ker V\) is closed densely defined symmetric in \(H\). When the singular perturbation \(V\) has some property of symmetry with respect to \(\eta\), the authors obtain a large family of nontrivial pseudo-Hermitian extensions of \(A_0\) and give necessary and sufficient conditions for the reality of their spectra. Moreover, a criterion for the similarity of a pseudo-Hermitian operator to a Hermitian one is obtained.
scientific article; zbMATH DE number 2115198