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

TL;DR

This paper advances the understanding of Landis' conjecture in the plane by establishing unique continuation estimates for solutions to elliptic equations with potentials that have exponentially decaying negative parts, using vanishing order estimates.

Contribution

It provides a new quantitative unique continuation result for elliptic equations with exponentially decaying negative potentials, extending previous work on Landis' conjecture.

Findings

Established unique continuation at infinity estimates for solutions with decaying negative potentials.

Proved an order of vanishing estimate using a similarity principle for Beltrami systems.

Applied an iteration scheme to derive the main theorem.

Abstract

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 non-trivial 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.