On the fractional Landis conjecture.

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 > 4 s / (4 s − 1) , then this solution must again be trivial. Remark that when s → 1 , 4 s / (4 s − 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. [ABSTRACT FROM AUTHOR]