On the Fractional Landis Conjecture

Abstract

In this paper we study a Landis-type conjecture for fractional Schrödinger 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.

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• Uniqueness and reconstruction for the fractional Calderón problem with a single measurement
• The fractional Calderón problem: low regularity and stability
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Cited by in corpus (1)

• Strong unique continuation for the higher order fractional Laplacian