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 $({\mathbb{C}}^n,{\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2)$ for $B^{\ast }$, the set of points in the spectrum of $A_1$ (the self-adjoint extension with domain $Ker{\mathrm{\Gamma }}_1$) which are not eigenvalues of maximum multiplicity for any self-adjoint extension of $B$ disjoint of $A_1$, is a dense ${\mathit{\text{G}}}_{\delta }$ 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.