Inequalities and separation for covariant Schrödinger operators

Abstract

We consider a differential expression LV∇=∇†∇+V, where ∇ is a metric covariant derivative on a Hermitian bundle E over a geodesically complete Riemannian manifold (M,g) with metric g, and V is a linear self-adjoint bundle map on E. In the language of Everitt and Giertz, the differential expression LV∇ is said to be separated in Lp(E) if for all u∈Lp(E) such that LV∇u∈Lp(E), we have Vu∈Lp(E). We give sufficient conditions for LV∇ to be separated in L2(E). We then study the problem of separation of LV∇ in the more general Lp-spaces, and give sufficient conditions for LV∇ to be separated in Lp(E), when 1<p<∞.