A multi-point maximum principle to prove global Harnack inequalities for Schr\"odinger 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 Schr\"odinger 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 $C^4$ 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 $C^2$ on $M$. In the particular case that $V$ is the quadratic potential $V(x)=|x|^2$ and $M$ is the Euclidean space $\mathbb{R}^d$, we prove a new Harnack inequality with sharper constants. Finally, we treat positive solutions of the Schr\"odinger equation with a gradient drift term, including applications to the Ornstein-Uhlenbeck operator $\Delta - x\cdot \nabla$ with quadratic potential in $\mathbb{R}^d$.