Boundary Conditions for Singular Perturbations of Self-Adjoint Operators

Let A : D(A) ⊆ H → H be an injective self-adjoint operator and let τ : D(A) → X, X a Banach space, be a surjective linear map such that ‖τφ‖X ≤ c ‖Aφ‖H. Supposing that Range (τ ) ∩ H = {0}, we define a family AτΘ of self-adjoint operators which are extensions of the symmetric operator A|{τ=0} . Any φ in the operator domain D(A τ Θ) is characterized by a sort of boundary conditions on its univocally defined regular component φreg, which belongs to the completion of D(A) w.r.t. the norm ‖Aφ‖H. These boundary conditions are written in terms of the map τ , playing the role of a trace (restriction) operator, as τφreg = ΘQφ, the extension parameter Θ being a self-adjoint operator from X to X. The self-adjoint extension is then simply defined by AτΘφ := Aφreg. The case in which Aφ = T ∗ φ is a convolution operator on L(R), T a distribution with compact support, is studied in detail.