Laplace-eigenfunctions on the torus with high vanishing order

TL;DR

This paper constructs Laplace eigenfunctions on high-dimensional tori with arbitrary vanishing orders using number theory, and applies these to address a question in spectral analysis and unique continuation.

Contribution

It introduces a novel method to create eigenfunctions with prescribed vanishing orders on tori, impacting the study of spectral projectors and Schrödinger operators.

Findings

Eigenfunctions with arbitrary vanishing orders are constructed.

Provides a negative answer to a question in spectral unique continuation.

Connects number theory with spectral analysis on tori.

Abstract

We use the sum-of-squares theorem from number theory to construct eigenfunctions of the Laplacian on the $d$-dimensional torus, $d \geq 2$, which vanish to any prescribed order at some point. These functions are then applied to provide a negative answer (in dimension $d \geq 2$) to a question in the context of quantitative unique continuation for spectral projectors of Schr\"odinger operators, asked by Egidi and Veseli\'c.