On the vanishing of eigenfunctions of the Laplacian on tori

TL;DR

This paper investigates the minimal local $L^2$-norm of Laplacian eigenfunctions on tori, using number theory and polynomial analysis, with implications for quantum limits and control theory.

Contribution

It offers new bounds on eigenfunction concentration on small regions of tori by combining geometric, algebraic, and analytical methods.

Findings

Partial bounds on eigenfunction $L^2$-norms on small balls

Connections established between eigenfunction behavior and integer points on spheres

Applications demonstrated in quantum limits and control theory

Abstract

Consider an eigenfunction of the Laplacian on a torus. How small can its $L^2$-norm be on small balls? We provide partial answers to this question by exploiting the distribution of integer points on spheres, basic properties of polynomials, and Nazarov--Tur\'an type estimates for exponential polynomials. Applications to quantum limits and control theory are given.