On Landis' conjecture in the plane for real-valued potentials with decay

TL;DR

This paper studies how real-valued solutions to planar Schrödinger equations with decaying potentials exhibit exponential decay, providing sharp estimates that depend explicitly on the decay rate of the potential.

Contribution

It extends previous results by establishing explicit decay estimates for solutions with potentials decaying at a rate N, using conformal transformations and iterative methods.

Findings

Exponential decay estimates depend explicitly on decay rate N

Estimates are essentially sharp for potentials with decay

Builds on and extends Landis' conjecture results in the plane

Abstract

We investigate the quantitative unique continuation properties of real-valued solutions to planar Schr\"odinger equations with potential functions that exhibit pointwise decay at infinity. That is, for equations of the form $-\Delta u + V u = 0$ in $\mathbb{R}^2$, where $|V(z)| \lesssim \langle z \rangle^{-N}$ for some $N > 0$, we prove that real-valued solutions satisfy exponential decay estimates with a rate that depends explicitly on $N$. Examples show that the estimates established here are essentially sharp. The case of $N = 0$ corresponds to the Landis conjecture, which was proved for real-valued solutions in the plane in [LMNN20], while the case of $N < 0$ was previously investigated by the author in [Dav24]. Here, the proof techniques rely on the ideas presented in [LMNN20] combined with conformal transformations and an iteration scheme.