Separation for the biharmonic differential operator in the Hilbert space associated with the existence and uniqueness theorem

In this paper, we have studied the separation for the following biharmonic differential operator: Au = u + V (x)u(x), x ∈ Rn, in the Hilbert space H = L2(Rn,H1) with the operator potential V (x) ∈ C1(Rn,L(H1)), where L(H1) is the space of all bounded linear operators on the Hilbert space H1 and u is the biharmonic differential operator, while u = n i=1 ∂2u ∂x2 i is the Laplace operator in Rn. Moreover, we have studied the existence and uniqueness of the solution of the biharmonic differential equation Au = u + V (x)u(x) = f (x) in the Hilbert space H, where f (x) ∈ H. © 2007 Elsevier Inc. All rights reserved.