Unique continuation for $\bar{\partial }u=Vu$ at infinity

Abstract

Motivated by Landis's conjecture on unique continuation at infinity for the Laplacian, we study the corresponding property for the Cauchy-Riemann operator. We prove that every weak solution of $\bar{\partial }u=Vu$ on a neighborhood of infinity, with $V\in L^{\infty }$, vanishes identically if it decays exponentially at a rate greater than $2\Vert V{\Vert }_{L^{\infty }}$. This conclusion is sharp both in the constant $2\Vert V{\Vert }_{L^{\infty }}$ and in the order of exponential decay required. More generally, we establish unique continuation at infinity for a broad class of radially decaying bounded potentials, with optimal decay rates determined by the decay of the potential. We also obtain related unique continuation results for $L^2$ potentials and for compactly supported potentials under weaker assumptions at infinity.

27 pages