Universal Constraints on the Location of Extrema of Eigenfunctions of Non-Local Schr\"odinger Operators

TL;DR

This paper establishes universal bounds on the location of eigenfunction extrema for non-local Schrödinger operators, revealing geometric and probabilistic implications and extending classical inequalities to non-local contexts.

Contribution

It introduces a universal lower bound on eigenfunction extrema location for non-local Schrödinger operators, incorporating the operator's symbol, potential strength, and eigenvalues, with broad geometric and probabilistic applications.

Findings

Derived a universal lower bound involving the operator's symbol and potential.

Extended Faber-Krahn inequality to non-local operators.

Provided bounds on extrema location relative to potential support and level sets.

Abstract

We derive a lower bound on the location of global extrema of eigenfunctions for a large class of non-local Schr\"odinger operators in convex domains under Dirichlet exterior conditions, featuring the symbol of the kinetic term, the strength of the potential, and the corresponding eigenvalue, and involving a new universal constant. We show a number of probabilistic and spectral geometric implications, and derive a Faber-Krahn type inequality for non-local operators. Our study also extends to potentials with compact support, and we establish bounds on the location of extrema relative to the boundary edge of the support or level sets around minima of the potential.