Separation property for Schrödinger operators in L p -spaces on non-compact manifolds

Abstract

Let (M, g) be a manifold of bounded geometry with metric g. We consider a Schrödinger-type differential expression L = Δ M + q, where Δ M is the scalar Laplacian on M and q is a non-negative locally integrable function on M. In the terminology of W.N. Everitt and M. Giertz, the differential expression L is said to be separated in L p (M) if for all u ∈ L p (M) such that Lu ∈ L p (M), we have qu ∈ L p (M). We give sufficient conditions for L to be separated in L p (M), where 1 < p < ∞.