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arXiv · 2506.21694

On Spectral Stability for Rank One Perturbations

Mario Alberto Ruiz Caballero·Rafael del Rio
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Abstract

Embedded point spectra of rank one singular perturbations of an arbitrary self-adjoint operator A on a Hilbert space H is studied. It is shown that 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 Gδ subset where it is not possible to have eigenvalues for any rank one singular perturbation. Moreover, for a dense Gδ set of rank one singular perturbations of A their eigenvalues are isolated. The approach presented here unifies points of view taken by different authors.

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Mario Alberto Ruiz Caballero, Rafael del Rio. 2025-06-26. On Spectral Stability for Rank One Perturbations. https://arxiv.org/abs/2506.21694

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