  Eigenfunctions of the fractional Schr\"odinger operators in a domain
$\mathcal{D}$ are considered, and a relation between the supremum of the
potential and the distance of a maximizer of the eigenfunction from
$\partial\mathcal{D}$ is established. This, in particular, extends a recent
result of Rachh and Steinerberger to the fractional Schr\"odinger operators. We
also propose a fractional version of the Barta's inequality and also generalize
a celebrated Lieb's theorem for fractional Schr\"odinger operators. As
applications of above results we obtain a Faber-Krahn inequality for non-local
Schr\"odinger operators.
