Singular perturbations of abstract wave equations

Given, on the Hilbert space H0, the self-adjoint operator B and the skew-adjoint operators C1 and C2, we consider, on the Hilbert space H D(B) ⊕H0, the skew-adjoint operator W =  C2 1 −B2 C1  corresponding to the abstract wave equation ¨ − (C1 + C2) ˙ =−(B2 + C1C2) . Given then an auxiliary Hilbert space h and a linear map : D(B2) → h with a kernel K dense in H0, we explicitly construct skew-adjoint operators W on a Hilbert space H D(B) ⊕H0 ⊕ h which coincide with W on N K⊕ D(B). The extension parameter ranges over the set of positive, bounded and injective self-adjoint operators on h. In the case C1 = C2 = 0 our construction allows a natural definition of negative (strongly) singular perturbations A of A := −B2 such that the diagram W −−−−−−→ W   A −−−−−−→ A is commutative. © 2005 Elsevier Inc. All rights reserved.