A multi-point maximum principle to prove global Harnack inequalities for Schrödinger operators

In this article, we introduce a new methodology to prove global parabolic Harnack inequalities on Riemannian manifolds. We focus on presenting a new proof of the global pointwise Harnack inequality satisfied by positive solutions of the linear Schrödinger equation on a Riemannian manifold M with nonnegative Ricci curvature, where the potential term V is bounded from below. Our approach is based on a multi-point maximum principle argument. Standard proofs of this result (see, for instance, Li-Yau [Acta Math, 1986]) rely on first establishing a gradient estimate. This requires the solution to be at least C⁴ C 4 on M . We instead prove the Harnack inequality directly, which has the advantage of avoiding higher-order derivatives of the solution in the proof, enabling us to assume it is only C² C 2 on M . In the particular case that V is the quadratic potential V(x)=|x|² V ( x ) = | x | 2 and M is the Euclidean space Rᵈ R d , we prove a new Harnack inequality with sharper constants. Finally, we treat positive solutions of the Schrödinger equation with a gradient drift term, including applications to the Ornstein-Uhlenbeck operator Δ - x· ∇ Δ - x · ∇ with quadratic potential in Rᵈ R d .