2025/06/26 by Mario Alberto Ruiz Caballero, Caballero, Mario Alberto Ruiz, Rafael del Rio +1
Mathematics · Computer Science · #Spectral Theory in Mathematical Physics #Holomorphic and Operator Theory #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2506.21694
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.