International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 38, Pages 2415-2423
doi:10.1155/S0161171203209212

On m-accretive Schrödinger-type operators with singular potentials on manifolds of bounded geometry

Abstract

We consider a Schrödinger-type differential expression ∇∗ ∇+V, where ∇ is a C∞-bounded Hermitian connection on a Hermitian vector bundle E of bounded geometry over a manifold of bounded geometry (M,g) with positive C∞-bounded measure dμ, and V is a locally integrable linear bundle endomorphism. We define a realization of ∇∗ ∇+V in L2(E) and give a sufficient condition for its m-accretiveness. The proof essentially follows the scheme of T. Kato, but it requires the use of a more general version of Kato's inequality for Bochner Laplacian operator as well as a result on the positivity of solution to a certain differential equation on M.