arXiv subjects

Mario Alberto Ruiz Caballero

Publications and source records attributed to Mario Alberto Ruiz Caballero.

2 recordsLinked to original sources

arXiv2026

On spectral stability for self-adjoint extensions

We prove that given a symmetric completely non-selfadjoint operator $B$ with finite deficiency indices $(n,n)$ on a Hilbert space and a boundary triplet $\left(\mathbb{C}^{n},Γ_{1},Γ_{2}\right)$ for $B^{*}$, the set of points in the spectrum of $A_{1}$ (the self-adjoint extension with domain $Ker\;Γ_{1}$) which are not eigenvalues of maximum multiplicity for any self-adjoint extension of $B$ disjoint of $A_{1}$, is a dense $\textit{G}_δ$ set in $σ(A_{1})$. Furthermore, a proof of a Malamud's theorem that generalizes a well-known result of the Aronszajn-Donoghue theory on the characterization of eigenvalues is offered.

math.SP↗

arXiv2025

On Spectral Stability for Rank One Perturbations

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.

math.SP↗