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) 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 p : E(h) P(h) , where P(h) denotes the set of orthogonal projections in the Hilbert space h D (A)/N and p -1 () is the set of self-adjoint operators in the range of . The set of self-adjoint operators in h , i.e. p -1 (1) , parametrises the relatively prime extensions. Any (, ) E(h) determines a boundary condition in the domain of the corresponding extension A , and explicitly appears in the formula for the resolvent (-A , + 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 Schrdinger operators with point interactions and to elliptic boundary value problems are given.