On spectral stability for self-adjoint extensions

TL;DR

This paper investigates the spectral stability of self-adjoint extensions of symmetric operators with finite deficiency indices, establishing dense sets of spectral points with specific eigenvalue properties and generalizing classical eigenvalue characterization theorems.

Contribution

It proves a dense G_delta set property for spectral points not being maximum multiplicity eigenvalues and generalizes Aronszajn-Donoghue eigenvalue characterization results.

Findings

Dense G_delta set of spectral points not eigenvalues of maximum multiplicity.

Generalization of Aronszajn-Donoghue eigenvalue characterization.

Spectral stability results for self-adjoint extensions with finite deficiency indices.

Abstract

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 $\left(\mathbb{C}^{n},\Gamma_{1},\Gamma_{2}\right)$ for $B^{*}$, the set of points in the spectrum of $A_{1}$ (the self-adjoint extension with domain $Ker\;\Gamma_{1}$) which are not eigenvalues of maximum multiplicity for any self-adjoint extension of $B$ disjoint of $A_{1}$, is a dense $\textit{G}_{\delta}$ set in $\sigma(A_{1})$. 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.