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 $\left(\mathbb{C}^{n},\Gamma_{1},\Gamma_{2}\right)$ 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.