Self-adjoint Extensions of Restrictions

We provide, by a resolvent Krĕın-like formula, all selfadjoint extensions of the symmetric operator S obtained by restricting the self-adjoint operator A : D(A) ⊆ H → H to the dense, closed with respect to the graph norm, subspace N ⊂ D(A). Neither the knowledge of S∗ nor of the deficiency spaces of S is required. Typically A is a differential operator and N is the kernel of some trace (restriction) operator along a null subset. We parametrise the extensions by the bundle π : E(h) → P(h), where P(h) denotes the set of orthogonal projections in the Hilbert space h ≃ D(A)/N and π(Π) is the set of self-adjoint operators in the range of Π. The set of self-adjoint operators in h, i.e. π(1), parametrises the relatively prime extensions. Any (Π,Θ) ∈ E(h) determines a boundary condition in the domain of the corresponding extension AΠ,Θ and explicitly appears in the formula for the resolvent (−AΠ,Θ + z). The connection with both von Neumann’s and Boundary Triples theories of self-adjoint extensions is explained. Some examples related to boundary value problems, to quantum graphs and to Schrödinger operators with point interactions are given.