On Landis' conjecture in the plane

In this paper we prove a quantitative form of Landis' conjecture in the plane. Precisely, let $W(z)$ be a measurable real vector-valued function and $V(z)\ge 0$ be a real measurable scalar function, satisfying $\|W\|_{L^{\infty}({\mathbf R}^2)}\le 1$ and $\|V\|_{L^{\infty}({\mathbf R}^2)}\le 1$. Let $u$ be a real solution of $\Delta u-\nabla(Wu)-Vu=0$ in ${\mathbf R}^2$. Assume that $u(0)=1$ and $|u(z)|\le\exp(C_0|z|)$. Then $u$ satisfies $\underset{{|z_0|=R}}{\inf}\,\underset{|z-z_0|<1}{\sup}|u(z)|\ge \exp(-CR\log R)$, where $C$ depends on $C_0$. In addition to the case of the whole plane, we also establish a quantitative form of Landis' conjecture defined in an exterior domain.