Unique continuation at infinity for potentials with arbitrary radial growth

Abstract

Let $G$ be any given continuous positive function on $\mathbb{R}_+$. Let $V$ be radial with $|V(x)|\leq G(|x|)$. We prove a Landis-type theorem for any real-valued solution of $\Delta u=Vu$ on $\mathbb{R}^n$. We construct a decay threshold $e^{-g(r)}$, where $g$ is a strictly increasing function which can be computed explicitly in terms of $G$. Under suitable assumptions the exponent in the decay threshold is proportional to the Agmon distance associated with $G$.