On Landis' conjecture in the plane for potentials with growth

Abstract

We investigate the quantitative unique continuation properties of real-valued solutions to Schr\"odinger equations in the plane with potentials that exhibit growth at infinity. More precisely, for equations of the form $\Delta u - V u = 0$ in $\mathbb{R}^2$, with $|V(z)| \lesssim |z|^{N}$ for some $N \ge 0$, we prove that real-valued solutions satisfy exponential decay estimates with a rate that depends explicitly on $N$. The case $N = 0$ corresponds to the Landis conjecture, which was proved for real-valued solutions in the plane in [LMNN20]. As such, the results in this article may be interpreted as generalized Landis-type theorems. Our proof techniques rely heavily on the ideas presented in [LMNN20].