The Landis conjecture on exponential decay

Abstract

Consider a solution $u$ to $\Delta u+Vu=0$ on ${\mathbb{R}}^2$, where $V$ is real-valued, measurable and $|V|\le 1$. If $|u(x)|\le \exp (-C|x|{\log }^{1/2}|x|)$, $|x|>2$, where $C$ is a sufficiently large absolute constant, then $u\equiv 0$.