Laplace-eigenfunctions on the torus with high vanishing order

Abstract

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