On the Fractional Landis Conjecture

TL;DR

This paper investigates a fractional Landis conjecture for fractional Schrödinger equations, establishing conditions under which solutions with specific decay rates must be trivial, extending classical results to fractional contexts.

Contribution

It provides new decay rate thresholds for triviality of solutions in fractional Schrödinger equations with differentiable and non-differentiable potentials, including quantitative estimates.

Findings

Solutions decay faster than certain exponential rates are trivial.

Derived decay thresholds depend on the fractional power s.

Quantitative estimates extend classical results to fractional cases.

Abstract

In this paper we study a Landis-type conjecture for fractional Schr\"odinger equations of fractional power $s\in(0,1)$ with potentials. We discuss both the cases of differentiable and non-differentiable potentials. On the one hand, it turns out for \emph{differentiable} potentials with some a priori bounds, if a solution decays at a rate $e^{-|x|^{1+}}$, then this solution is trivial. On the other hand, for $s\in(1/4,1)$ and merely bounded \emph{non-differentiable} potentials, if a solution decays at a rate $e^{-|x|^\alpha}$ with $\alpha>4s/(4s-1)$, then this solution must again be trivial. Remark that when $s\to 1$, $4s/(4s-1)\to 4/3$ which is the optimal exponent for the standard Laplacian. For the case of non-differential potentials and $s\in(1/4,1)$, we also derive a quantitative estimate mimicking the classical result by Bourgain and Kenig.