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

TL;DR

This paper investigates the location of eigenfunction maximizers for fractional Schr"odinger operators, establishing new bounds and inequalities that extend classical results to the non-local fractional setting.

Contribution

It introduces a fractional version of Barta's inequality, generalizes Lieb's theorem, and extends the Faber-Krahn inequality to non-local Schr"odinger operators.

Findings

Relation between potential supremum and eigenfunction maximizer distance

Fractional Barta's inequality established

Generalized Lieb's theorem for fractional operators

Abstract

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.