Self-adjoint Extensions by Additive Perturbations

Let AN be the symmetric operator given by the restriction of A toN , 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 AΘ of AN such that D(AΘ)∩D(A) = N can be additively decomposed by the sum AΘ = Ā + TΘ, where both the operators Ā and TΘ take values in the strong dual of D(A). The operator Ā is the closed extension of A to the whole H whereas TΘ 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 AN . The explicit connection with both Krĕın’s resolvent formula and von Neumann’s theory of self-adjoint extensions is given.