TY  - JOUR
ID  - Caballero2025
LB  - Caballero2025
AU  - Caballero, Mario Alberto Ruiz
AU  - Río, Rafael del
TI  - On spectral stability for rank one singular perturbations
ST  - On spectral stability for rank one singular perturbations
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2025
VL  - 45
IS  - 6
SP  - 819
EP  - 840
DO  - https://doi.org/10.7494/OpMath.2025.45.6.819
UR  - https://doi.org/10.7494/OpMath.2025.45.6.819
AB  - 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.
KW  - self-adjoint extension
KW  - rank one singular perturbation
KW  - embedded point spectra
KW  - singular continuous spectrum
SN  - 1232-9274
ER  -