Unitary groups and spectral sets

TL;DR

This paper explores the spectral properties of unions of intervals in one dimension, linking geometric configurations to spectral theory through selfadjoint operators and unitary groups, with implications for the Fuglede conjecture.

Contribution

It provides a detailed characterization of spectral sets in terms of selfadjoint extensions and unitary representations, connecting geometry with spectral properties for unions of intervals.

Findings

Characterization of spectral sets via selfadjoint extensions

Explicit link between geometry of intervals and Fourier spectra

Reduction of Fuglede conjecture to unions of integer intervals

Abstract

We study spectral theory for bounded Borel subsets of $\br$ and in particular finite unions of intervals. For Hilbert space, we take $L^2$ of the union of the intervals. This yields a boundary value problem arising from the minimal operator $\Ds = \frac1{2\pi i}\frac{d}{dx}$ with domain consisting of $C^\infty$ functions vanishing at the endpoints. We offer a detailed interplay between geometric configurations of unions of intervals and a spectral theory for the corresponding selfadjoint extensions of $\Ds$ and for the associated unitary groups of local translations. While motivated by scattering theory and quantum graphs, our present focus is on the Fuglede-spectral pair problem. Stated more generally, this problem asks for a determination of those bounded Borel sets $\Omega$ in $\br^k$ such that $L^2(\Omega)$ has an orthogonal basis of Fourier frequencies (spectrum), i.e., a total set of orthogonal complex exponentials restricted to $\Omega$. In the general case, we characterize Borel sets $\Omega$ having this spectral property in terms of a unitary representation of $(\br, +)$ acting by local translations. The case of $k = 1$ is of special interest, hence the interval-configurations. We give a characterization of those geometric interval-configurations which allow Fourier spectra directly in terms of the selfadjoint extensions of the minimal operator $\Ds$. This allows for a direct and explicit interplay between geometry and spectra. As an application, we offer a new look at the Universal Tiling Conjecture and show that the spectral-implies-tile part of the Fuglede conjecture is equivalent to it and can be reduced to a variant of the Fuglede conjecture for unions of integer intervals.