Location of Maximizers of Eigenfunctions of Fractional Schrödinger’s Equations

Abstract

Eigenfunctions of the fractional Schrödinger operators in a domain D are considered, and a relation between the supremum of the potential and the distance of a maximizer of the eigenfunction from ∂ D is established. This, in particular, extends a recent result of Rachh and Steinerberger arXiv:1608.06604 (2017) to the fractional Schrödinger operators. We also propose a fractional version of the Barta’s inequality and also generalize a celebrated Lieb’s theorem for fractional Schrödinger operators. As applications of above results we obtain a Faber-Krahn inequality for non-local Schrödinger operators.