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 √det g(x) ∂ ∂xi det g(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 − 1 √det g(x) ∂ ∂xi det g(x)g−1(x) ∂u ∂xj + V (x)u(x) = f (x), f(x) ∈ H, in the Hilbert space H = L2(Ω,H1). © 2007 Elsevier Inc. All rights reserved