Self-adjoint extensions by additive perturbations

Posilicano, Andrea

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 1-20.

Let $A_{𝒩}$ be the symmetric operator given by the restriction of $A$ to $𝒩$, where $A$ is a self-adjoint operator on the Hilbert space $\mathscr{H}$ and $𝒩$ is a linear dense set which is closed with respect to the graph norm on $D\left(A\right)$, the operator domain of $A$. We show that any self-adjoint extension $A_{\Theta }$ of $A_{𝒩}$ such that $D\left(A_{\Theta }\right)\cap D\left(A\right)=𝒩$ can be additively decomposed by the sum $A_{\Theta }=\overline{A}+T_{\Theta }$, where both the operators $\overline{A}$ and $T_{\Theta }$ take values in the strong dual of $D\left(A\right)$. The operator $\overline{A}$ is the closed extension of $A$ to the whole $\mathscr{H}$ whereas $T_{\Theta }$ is explicitly written in terms of a (abstract) boundary condition depending on $𝒩$ and on the extension parameter $\Theta$, a self-adjoint operator on an auxiliary Hilbert space isomorphic (as a set) to the deficiency spaces of $A_{𝒩}$. The explicit connection with both Kreĭn’s resolvent formula and von Neumann’s theory of self-adjoint extensions is given.

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