SELF-ADJOINT EXTENSIONS OF RESTRICTIONS

We provide a simple recipe for obtaining all self-adjoint extensions, together with their resolvent, of the symmetric operator S obtained by restricting the self-adjoint operator A : D (A) subset of H -> H to the dense, closed with respect to the graph norm, subspace N subset of 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 p : E(h) -> P(h), where P(h) denotes the set of orthogonal projections in the Hilbert space h similar or equal to D (A)/N and p(-1)(II) is the set of self-adjoint operators in the range of Pi. The set of self-adjoint operators in h, i.e. p(-1)(1), parametrises the relatively prime extensions. Any (Pi, Theta) is an element of E(h) determines a boundary condition in the domain of the corresponding extension A(Pi,Theta) and explicitly appears in the formula for the resolvent (-A(Pi,Theta) + z)(-1). The connection with both von Neumann's and Boundary Triples theories of self-adjoint extensions is explained. Some examples related to quantum graphs, to Schrodinger operators with point interactions and to elliptic boundary value problems are given.