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

Abstract

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).