On spectral stability for rank one singular perturbations

Abstract

We study the embedded point spectrum of rank one singular perturbations of an arbitrary self-adjoint operator $A$ on a Hilbert space $\mathcal{H}$. These perturbations can be regarded as self-adjoint extensions of a densely defined closed symmetric operator $B$ with deficiency indices $(1,1)$. Assuming the deficiency vector of $B$ is cyclic for its self-adjoint extensions, we prove that the spectrum of $A$ contains a dense $G_{\delta}$ subset on which no eigenvalues occur for the rank one singular perturbations considered. We show this is equivalent to the existence of a dense $G_{\delta}$ set of rank one singular perturbations of $A$ such that their eigenvalues are isolated. The approach presented here unifies points of view taken by different authors.