Mass non-concentration at the nodal set and a sharp Wasserstein uncertainty principle

TL;DR

This paper establishes mass concentration properties of Laplace eigenfunctions away from their nodal sets across all dimensions and derives a sharp Wasserstein uncertainty principle that confirms a previous conjecture.

Contribution

It extends previous results on eigenfunction mass concentration to all dimensions and introduces a sharp Wasserstein uncertainty principle valid in high-frequency regimes.

Findings

Mass concentration of eigenfunctions away from nodal sets in all dimensions

A sharp Wasserstein uncertainty principle proven in the high-frequency limit

Confirmation of a conjecture in the literature

Abstract

We prove $L^p$-mass concentration properties of Laplace eigenfunctions away from their nodal sets, extending a recent result in \cite{GM3} to all dimensions, and giving a slight refinement of a result in \cite{JN}. As a consequence, we are able to derive a sharp Wasserstein uncertainty principle that holds uniformly in the high frequency regime, proving a conjecture in \cite{St1}.