On the fractional Landis conjecture - Rüland, Angkana; Wang, Jenn-Nan (Journal of functional analysis, 2019)
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-differentiable potentials and s∈(1/4,1), we also derive a quantitative estimate mimicking the classical result by Bourgain and Kenig.