Cyclicity in rank-1 perturbation problems (Q2854253)

The authors study the property of cyclicity for rank-1 perturbations of self-adjoint and unitary operators in a Hilbert space \(\mathcal{H}\). For \(A\) a self-adjoint operator in \(\mathcal{H}\) and \(\varphi\) a cyclic vector for \(A\), the authors consider the family of rank-1 perturbations \(A_{\alpha} = A+\alpha(\cdot,\varphi)\varphi\), \(\alpha \in \mathbb{R}\), and prove that, if \(0\not= f \in \mathcal{H}\), then \(f\) is a cyclic vector for \((A_{\alpha})_{ac}\) for all but a countable number of \(\alpha \in \mathbb{R}\) and \(f\) is a cyclic vector for \((A_{\alpha})_{s}\) for Lebesgue a.e. \(\alpha \in \mathbb{R}\). Here, \((A_{\alpha})_{ac}\) and \((A_{\alpha})_{s}\) denote the absolutely continuous part and the singular part of the operator \(A_{\alpha}\), respectively. An application of this result to Anderson-type Hamiltonians is given (related to this application, see also [\textit{V. Jakšić} and \textit{Y. Last}, Duke Math. J. 133, No. 1, 185--204 (2006; Zbl 1107.47027)]). In the case of unitary operators, the authors prove that, if \(U_{\gamma}=U+(\gamma-1)(\cdot,U^{-1}b)b\), \(\gamma \in \mathbb{T}\), for some cyclic vector \(b\) for \(U\), \(\|b\|=1\), and if \(c \in \mathbb{C} \setminus \{0\}\), then the vector \(f-cb\) is cyclic for \(U_{\gamma}\) for all \(\gamma \in \mathbb{T} \setminus \{ e^{2i \mathrm{arg} c}\}\) and all \(0\not= f \in \mathcal{H}\) Hermitian with respect to \(U\) and \(b\).