Inequalities and separation for the Laplace-Beltrami differential operator in Hilbert spaces

In this paper we have studied the separation for the Laplace-Beltrami differential operator of the formAu=−1detg(x)∂∂xi[detg(x)g−1(x)∂u∂xj]+V(x)u(x),∀x=(x1,x2,…,xn)∈Ω⊂Rn, in the Hilbert space H=L2(Ω,H1), with the operator potential V(x)∈C1(Ω,L(H1)), where L(H1) is the space of all bounded linear operators on the arbitrary Hilbert space H1 and g(x)=(gij(x)) is the Riemannian matrix, while g−1(x) is the inverse of the matrix g(x). Also we have studied the existence and uniqueness of the solution for the Laplace-Beltrami differential equation of the form−1detg(x)∂∂xi[detg(x)g−1(x)∂u∂xj]+V(x)u(x)=f(x),f(x)∈H, in the Hilbert space H=L2(Ω,H1).