On the vanishing of eigenfunctions of the Laplacian on tori

2024/06/28 by Germain, Pierre, Moyano, Iván, Zhu, Hui · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2406.19925

Abstract

Consider an eigenfunction of the Laplacian on a torus. How small can its L²-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.

Cited by

• Observability of Schrödinger propagators on tori in rough settings

Related

• Restriction of toral eigenfunctions to hypersurfaces and nodal sets
• Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus
• Eigenfunction maxima and spherical means
• Quantitative equidistribution properties of toral eigenfunctions
• L^p norms, nodal sets, and quantum ergodicity