Spectrality of product domains and Fuglede's conjecture for convex polytopes

Abstract

A set $\Omega \subset {\mathbb{R}}^d$ is said to be spectral if the space $L^2(\Omega )$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets "behave like" sets which can tile the space by translations. This suggests a conjecture that a product set $\Omega =A\times B$ is spectral if and only if the factors $A$ and $B$ are both spectral sets. We recently proved this in the case when $A$ is an interval in dimension one. The main result of the present paper is that the conjecture is true also when $A$ is a convex polygon in two dimensions. We discuss this result in connection with the conjecture that a convex polytope $\Omega$ is spectral if and only if it can tile by translations.

To appear in Journal d'Analyse Mathematique