Location of maximizers of eigenfunctions of fractional Schr\"odinger's equation

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.