Unique continuation at infinity for Schr\"odinger equations with Reverse H\"older Potentials

Abstract

In this article, we study unique continuation properties at infinity for solutions to generalized Schr\"odinger equations with potential functions that belong to the reverse H\"older class. For equations of the form $-div (A \nabla u) + V u = 0$ in $\mathbb{R}^n$, where $A$ is bounded and elliptic and $V \in RH_p$ for some $p \in [\frac n 2, \infty]$, we prove that if a solution doesn't grow too quickly at infinity, then it must be trivial. We use $d_V$, the Agmon distance function associated to $V$, to quantify the threshold growth rate. More precisely, there exists a constant $\gamma_0>0$ so that if $|u(x)| \lesssim \exp(\gamma d_V(x, 0))$ for some $\gamma<\gamma_0$ and every $x \in \mathbb{R}^n$, then $u$ must be trivial. The result may be interpreted as a Liouville-type theorem and is related to Landis'conjecture. Our proof techniques are inspired by Z. Shen's exponential decay estimates for fundamental solutions of Schr\"odinger operators and involve the application of a Fefferman-Phong inequality.