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 operatorG=G0+V(x),x∈Rn,in the weighted Hilbert space H=L2,k(Rn,H1), where G0=−∑i,j=1naij(x)Dij is the differential operator with the real positive coefficients aij(x)∈C2(Rn) and Dij=∂2∂xi∂xj,i,j=1,…,n. The operator potential V(x)∈C1(Rn,L(H1)), where L(H1) is the space of all bounded linear operators on the arbitrary Hilbert space H1. Moreover, we study the existence and uniqueness of the solution of the second order differential equation −∑i,j=1naij(x)Diju(x)+V(x)u(x)=f(x), where f(x)∈H, in the weighted Hilbert space H=L2,k(Rn,H1).