@Article{Caballero2025,
author = "Caballero, Mario Alberto Ruiz and Río, Rafael del",
title = "On spectral stability for rank one singular perturbations",
journal = "Opuscula Math.",
year = "2025",
volume = "45",
number = "6",
pages = "819--840",
doi = "https://doi.org/10.7494/OpMath.2025.45.6.819",
url = "https://doi.org/10.7494/OpMath.2025.45.6.819",
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.",
keywords = "self-adjoint extension, rank one singular perturbation, embedded point spectra, singular continuous spectrum",
issn = "1232-9274"
}