Self-Adjoint Extensions by Additive Perturbations

Abstract

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 $\H$ and $\N$ 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 $A_Θ$ of $A_\N$ such that $D(A_Θ)\cap D(A)=\N$ can be additively decomposed by the sum $A_Θ=\A+T_Θ$, where both the operators $\A$ and $T_Θ$ take values in the strong dual of $D(A)$. The operator $\A$ is the closed extension of $A$ to the whole $\H$ whereas $T_Θ$ is explicitly written in terms of a (abstract) boundary condition depending on $\N$ and on the extension parameter $Θ$, 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 Kre\uın's resolvent formula and von Neumann's theory of self-adjoint extensions is given.