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 form Au = -1/root det g(x) partial derivative/partial derivative x(i) {[}root det g(x) g(-1)(x) partial derivative u/partial derivative x(j)] + V(x)u(x), for all x = (x(1), x(2), . . . , x(n)) epsilon Omega subset of R-n , in the Hilbert space H = L2(Omega, HI), with the operator potential V(x) E C-1(Omega, L(H-1)), where L(Hi) is the space of all bounded linear operators on the arbitrary Hilbert space HI and g(x) = (g(ij) (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 -1/root det g(x) partial derivative/partial derivative x(i) {[}root det g(x) g(-1)(x) partial derivative u/partial derivative x(j)] + V(x)u(x) = f(x) epsilon H, in the Hilbert space H = L2(Omega, HI). (c) 2007 Elsevier Inc. All rights reserved.