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 Schrödinger equations with potential functions that exhibit pointwise decay at infinity. That is, for equations of the form $-\Delta u+Vu=0$ in ${\mathbb{R}}^2$, where $|V(z)|\lesssim ⟨z{⟩}^{-N}$ 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.

arXiv comment16 pages, 2 figures