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

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.