Fuglede's conjecture, differential operators and unitary groups of local translations

TL;DR

This paper explores the connections between Fourier spectra, differential operators, and domain geometry in the context of Fuglede's conjecture, highlighting both historical and recent developments in $L^2$ spaces and translation-tiling systems.

Contribution

It provides a comprehensive analysis of the interplay between spectral sets, differential operators, and geometric tilings, extending Fuglede's original question with new insights and results.

Findings

Characterization of orthogonal Fourier bases on various domains

Extensions of partial derivative operators linked to spectral properties

Identification of geometric conditions for translation-tiling domains

Abstract

The purpose of the present paper is to address multiple aspects of the Fuglede question dealing (Fourier spectra vs geometry) with a variety of $L^2$ contexts where we make precise the interplay between the three sides of the question: (i) existence of orthogonal families of Fourier basis functions (and associated spectra) on the one hand, (ii) extensions of partial derivative operators, and (iii) geometry of the corresponding domains, stressing systems of translation-tiles. We emphasize an account of old and new developments since the original 1974-paper by Bent Fuglede where the co-authors and Steen Pedersen have contributed.