Coexistence of spectra in rank-one perturbation problems.
Let \(A_0\) be a selfadjoint operator in a separable Hilbert space. Let \(A_\lambda= A_0+\lambda(.,\varphi)\varphi\), \(\lambda\in\mathbb{R}\). The following result is proven: For any measurable set \(B\subset\mathbb{R}\) there exists a family of rank one perturbations \(\{A_\lambda\}\) such that \(A_\lambda\) has dense absolutely continuous and dense singular spectrum for almost every \(\lambda\in B\) and dense absolutely continuous (but no singular) spectrum for almost every \(\lambda\not\in B\).
scientific article; zbMATH DE number 1809797