Singular perturbations of abstract wave equations

摘要

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 similar or equal to D(B) circle plus H-0, the skew-adjoint operator[GRAPHICS]corresponding to the abstract wave equation phi - (C-1 + C-2)phi = - (B-2 + C1C2)phi. Given then an auxiliary Hilbert space h and a linear map tau : D(B-2) -> h with a kernel H dense in H-0, we explicitly construct skew-adjoint operators W-Theta on a Hilbert space H-Theta similar or equal to D(B) circle plus H-0 circle plus h which coincide with W on N similar or equal to K circle plus D D(B). The extension parameter Theta 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(Theta) of A := -B-2 such that the diagram[GRAPHICS]is commutative. (c) 2005 Elsevier Inc. All rights reserved.