Rank-one singular perturbations with a dual pair of eigenvalues
Let \(A\) be a self-adjoint unbounded operator in a separable Hilbert space~\(H\) with dense domain \(D(A)\) and a nonempty connected spectral gap (i.e., the resolvent set \(\rho(A)\) of \(A\) has the property that the set \(\rho(A)\cap {\mathbb R}\) is connected). Let \(\{H_k\}_{k\in{\mathbb R}}\) denote the associated \(A\)-scale of Hilbert spaces (\(H_2=D(A)\)) and \(\langle \cdot,\cdot \rangle\) the dual inner product between \(H_k\) and \(H_{-k}\). The authors consider the class~\(P_s^1(A)\) of self-adjoint operators~\(\widetilde A\) obtained by rank-one singular perturbations of~\(A\). Each \(\widetilde A\in P_s^1(A)\) admits the representation \(\widetilde A = A {\widetilde+} \alpha\langle \cdot,\omega\rangle\omega\), where \(0\neq \alpha\in {\mathbb R}\), \(\omega\in H_{-2}\), and \(\widetilde+\) stands for the generalized sum of operators. A new phenomenon is studied: a rank-one singular perturbation may produce appearance of a dual pair of eigenvalues, one in the resolvent set \(\rho(A)\) and the other in the spectrum \(\sigma(A)\) of \(A\). Here, for a fixed vector \(\psi\in H\setminus D(A)\), \(\| \psi\| =1\), a point \(\lambda\in\sigma(A)\) is said to be dual with respect to a given \(\mu\in\rho(A)\) if \((\lambda-\mu)^{-1}=(\psi,R_\mu\psi)\), where \(R_\mu:=(A-\mu)^{-1}\). The main result of the present paper is the following. For any \(\psi\in H\setminus D(A)\), \(\| \psi\| =1\), and any real \(\mu\in\rho(A)\) there exists a unique \(\widetilde A\in P_s^1(A)\) such that both \(\mu\) and its dual point \(\lambda\in \sigma(A)\) with respect to \(\psi\) are eigenvalues of \(\widetilde A\); more exactly, with \(\phi:=(A-\lambda)R_\mu\psi\), there holds \(\widetilde A \psi = \lambda \psi\), \(\widetilde A \phi = \mu \phi\). The key role in the proof is played by the well-known Kreĭn formula for the resolvent of \(\widetilde A\).
scientific article; zbMATH DE number 1967854