Small scale equidistribution of eigenfunctions on the torus

TL;DR

This paper investigates how eigenfunctions of the Laplacian distribute their $L^2$ mass at small scales on flat tori, establishing equidistribution results for most eigenfunctions and constructing examples of irregularities.

Contribution

It proves small scale equidistribution for a density one subsequence of eigenfunctions on flat tori, including down to the Planck scale in dimension two, and constructs eigenfunctions with irregular mass distribution.

Findings

Existence of a density one subsequence with small scale equidistribution.

Construction of eigenfunctions with non-equidistributing $L^2$ mass.

Small scale equidistribution results in dimensions 2, 3, and 4.

Abstract

We study the small scale distribution of the $L^2$ mass of eigenfunctions of the Laplacian on the flat torus $\mathbb T^d$. Given an orthonormal basis of eigenfunctions, we show the existence of a density one subsequence whose $L^2$ mass equidistributes at small scales. In dimension two our result holds all the way down to the Planck scale. For dimensions $d=3,4$ we can restrict to individual eigenspaces and show small scale equidistribution in that context. We also study irregularities of quantum equidistribution: We construct eigenfunctions whose $L^2$ mass does not equidistribute at all scales above the Planck scale. Additionally, in dimension $d=4$ we show the existence of eigenfunctions for which the proportion of $L^2$ mass in small balls blows up at certain scales.