Separation property for Schrödinger operators on Riemannian manifolds

We consider a Schrödinger differential expression L=ΔM+q on a complete Riemannian manifold (M,g)(M,g) with metric g  , where ΔM is the scalar Laplacian on M   and q≥0q≥0 is a locally square integrable function on M. In the terminology of Everitt and Giertz, the differential expression L   is said to be separated in L2(M)L2(M) if for all u∈L2(M)u∈L2(M) such that Lu∈L2(M)Lu∈L2(M), we have qu∈L2(M)qu∈L2(M). We give sufficient conditions for L   to be separated in L2(M)L2(M).