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

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The purpose of this paper is to study the separation for the general second order elliptic differential operator G=G 0+V(x),x∈R n, in the weighted Hilbert space H= L 2, k ( R n , H 1), where G 0=−∑ i, j=1 n a ij ( x) D i j is the differential operator with the real positive coefficients a ij ( x)∈ C 2( R n ) and D i j= ∂ 2 ∂x i ∂x j ,i,j=1,…,n . The operator potential V( x)∈ 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 −∑ i, j=1 n a ij ( x) D i j u( x)+ V( x) u( x)= f( x), where f( x)∈ H, in the weighted Hilbert space H= L 2, k ( R n , H 1).