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 = Delta Delta u + V(x)u(x), x epsilon R-n, in the Hilbert space H = L-2(R-n, H-1) with the operator potential V(x) epsilon C-1(R-n, L(H-1)), where L(H-1) is the space of all bounded linear operators on the Hilbert space H-1 and Delta Delta u is the biharmonic differential operator, while Delta u = Sigma(n)(i=1) partial derivative(2)u/partial derivative x(i)(2) is the Laplace operator in R-n. Moreover, we have studied the existence and uniqueness of the solution of the biharmonic differential equation Au = Delta Delta u + V(x)u(x) = f(x) in the Hilbert space H, where f (x) epsilon H. (C) 2007 Elsevier Inc. All rights reserved.