# On Landis Conjecture for the Fractional Schr\"{o}dinger Equation

## TL;DR

This paper investigates a Landis-type conjecture for the fractional Schrödinger equation, establishing decay conditions under which solutions must be trivial, and extends previous work with new Carleman estimates for non-smooth potentials.

## Contribution

It proves decay-based uniqueness results for fractional Schrödinger equations with both smooth and non-smooth potentials, extending prior results and analyzing the operator's properties.

## Key findings

- Solutions with rapid decay are identically zero for smooth potentials.
- Solutions with specific decay rates are trivial for non-smooth potentials.
- The operator $(-P)^s$ is additive and bounded for non-smooth coefficients.

## Abstract

In this paper, we study a Landis-type conjecture for the general fractional Schr\"{o}dinger equation $((-P)^{s}+q)u=0$. As a byproduct, we also proved the additivity and boundedness of the linear operator $(-P)^{s}$ for non-smooth coefficents. For differentiable potentials $q$, if a solution decays at a rate $\exp(-|x|^{1+})$, then the solution vanishes identically. For non-differentiable potentials $q$, if a solution decays at a rate $\exp(-|x|^{\frac{4s}{4s-1}+})$, then the solution must again be trivial. The proof relies on delicate Carleman estimates. This study is an extension of the work by R\"{u}land-Wang (2019).