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Eigenfunctions of the fractional Schr\"{o}dinger operators in a domain D will be considered, and a relation between the supremum of the potential and the distance of a extrema of the eigenfunction from $\partial{D}$ will be established. This results, in particular, extends a recent result of Rachh and Steinerberger (2017), to the fractional Schr\"{o}dinger operators. We also generalize a celebrated Lieb’s theorem for fractional Schr\"{o}dinger operators. As an application of these results we obtain a Faber-Krahn inequality for non-local Schr\"{o}dinger operators. Extensions to more general non-local operators will also be discussed.
This is based on a joint work with J\"{o}zsef L\H{o}rinczi. |