On Landis' conjecture in the plane when the potential has an exponentially decaying negative part

In this article, we continue our investigation into the unique continuation properties of real-valued solutions to elliptic equations in the plane. More precisely, we make another step towards proving a quantitative version of Landis' conjecture by establishing unique continuation at infinity estimates for solutions to equations of the form $ - \Delta u + V u = 0$ in $ \mathbb{R}^2$, where $ V = V_+ - V_-$, $ V_+ \in L^\infty $, and $ V_-$ is a nontrivial function that exhibits exponential decay at infinity. The main tool in the proof of this theorem is an order of vanishing estimate in combination with an iteration scheme. To prove the order of vanishing estimate, we establish a similarity principle for vector-valued Beltrami systems.