Self-Adjoint Extensions by Additive Perturbations

Abstract

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

Revised version. To appear in: Ann. Scuola Norm. Sup. Pisa Cl. Sci

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