On spectral stability for rank one singular perturbations

Mario Alberto Ruiz Caballero
Rafael del Río

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.

Mathematics Subject Classification: 47B02, 47B25, 47A55, 47A10.

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