Singular perturbations of abstract wave equations

Abstract

Given, on the Hilbert space H 0 , the self-adjoint operator B and the skew-adjoint operators C 1 and C 2 , we consider, on the Hilbert space H ≃ D ( B ) ⊕ H 0 , the skew-adjoint operator W = C 2 1 - B 2 C 1 corresponding to the abstract wave equation φ ¨ - ( C 1 + C 2 ) φ ˙ = - ( B 2 + C 1 C 2 ) φ . Given then an auxiliary Hilbert space h and a linear map τ : D ( B 2 ) → h with a kernel K dense in H 0 , we explicitly construct skew-adjoint operators W Θ on a Hilbert space H Θ ≃ D ( B ) ⊕ H 0 ⊕ h which coincide with W on N ≃ K ⊕ D ( B ) . The extension parameter Θ ranges over the set of positive, bounded and injective self-adjoint operators on h . In the case C 1 = C 2 = 0 our construction allows a natural definition of negative (strongly) singular perturbations A Θ of A ≔ - B 2 such that the diagram W → W Θ A → A Θ is commutative.