On the Fractional Landis Conjecture

2018/09/12 by Rüland, Angkana, Wang, Jenn-Nan · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1809.04480

Abstract

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

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