Boundary conditions for singular perturbations of self-adjoint operators

Let A : D(A) subset of or equal to H --> H be an injective self-adjoint operator and let tau : D(A) --> X, X a Banach space, be a surjective linear map such that \\tauphi\\x less than or equal to c\\Aphi\\(H) Supposing that Kernel tau is dense in H, we define a family A(Theta)(tau) of sclf-adjoint operators which are extensions of the symmetric operator A(\{tau=0}). Any phi in the operator domain D(A(Theta)(tau)) is characterized by a sort of boundary conditions on its univocally defined regular component phi(reg) which belongs to the completion of D(A) w.r.t. tile norm \\Aphi\\(H). These boundary conditions arc written in terms of the map tau, playing the role of a trace (restriction) operator, as tauphi(reg) + ThetaQ(phi) the extension parameter Theta being a self-adjoint operator from X' to X. The self-adjoint extension is then simply defined by A(Theta)(tau)phi := Aphi(reg). The case in which Aphi = Psi * phi is a convolution operator on L-2 (R-n), Psi a distribution with compact support, is studied in detail.