  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.
