Semiclassical limits of eigenfunctions on flat $n$-dimensional tori

TL;DR

This paper proves a conjecture that the Fourier transform of squared eigenfunctions on flat n-dimensional tori has uniform bounds independent of eigenvalues, extending previous results to higher dimensions and implications for semiclassical limits.

Contribution

It generalizes Jakobson's argument to n-dimensional tori, establishing uniform bounds for Fourier transforms of eigenfunction squares and semiclassical limits.

Findings

Uniform $l^n$ bounds for Fourier transforms of eigenfunction squares

Extension of bounds to higher-dimensional tori

A geometric lemma bounding certain simplices on spheres

Abstract

We provide a proof of the conjecture formulated in \cite{Jak97,JNT01} which states that on a $n$-dimensional flat torus $\T^{n}$, the Fourier transform of squares of the eigenfunctions $|\phi_\lambda|^2$ of the Laplacian have uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof is a generalization of the argument by Jakobson, {\it et al}. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on $\TT^{n+2}$. We also prove a geometric lemma that bounds the number of codimension-one simplices which satisfy a certain restriction on an $n$-dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$ and use it in the proof.