Separation of the general second order elliptic differential operator with an operator potential in the weighted Hilbert spaces

The purpose of this paper is to study the separation for the general second order elliptic differential operator G = G(0) + V(x),x is an element of R-n, in the weighted Hilbert space H = L-2,L-k (R-n, H-1), where G(0) = -Sigma(i.j=1)(n)a(ij)(x)D-i(j) is the differential operator with the real positive coefficient, a(ij)(x) is an element of C-2(R-n) and D-i(j) = partial derivative(2)/partial derivative(xi)partial derivative(xj), i, j = 1, ..., n. The operator potential V(x) is an element of C-1(R-n, L(H-1)), where L(H-1) is the space of all bounded linear operators on the arbitrary Hilbert space H-1. Moreover, we study the existence and uniqueness of the solution of the second order differential equation -Sigma(i.j=1)(n) a(ij)(x)D(i)(j)u(x) + V(x)u(x) = f(x), where f(x) is an element of H, in the weighted Hilbert space H = L-2,L-k (R-n, H-1). (C) 2004 Published by Elsevier Inc.