On the vanishing of eigenfunctions of the Laplacian on tori

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--Turán type estimates for exponential polynom...

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Bibliographic Details
Published in:arXiv.org 2024-09
Main Authors: Germain, Pierre, Moyano, Iván, Zhu, Hui
Format: Article
Language:English
Subjects:
Control theory
Eigenvectors
Polynomials
Toruses
Online Access:Get full text
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Summary: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--Turán type estimates for exponential polynomials. Applications to quantum limits and control theory are given.
ISSN:2331-8422