Boundary Conditions for Singular Perturbations

Let A : D(A) H ! H be an injective self-adjoint operator and let : D(A) ! X, X a Banach space, be a surjective linear map such that k k X c kAAk H. Supposing that Range (0) \ H 0 = f0g, we deene a family A of self-adjoint operators which are extensions of the symmetric operator A jf =0g. Any in the operator domain D(A) is characterized by a sort of boundary conditions on its univocally deened regular component reg , which belongs to the completion of D(A) w.r.t. the norm kAAk H. These boundary conditions are written in terms of the map , playing the role of a trace (restriction) operator, as reg = Q , the extension parameter being a self-adjoint operator from X 0 to X. The self-adjoint extension is then simply deened by A := A reg. The case in which AA = T is a convolution operator on L 2 (R n), T a distribution with compact support, is studied in detail.