Spectral properties of unions of intervals and groups of local translations

TL;DR

This paper investigates the spectral properties of unions of intervals by analyzing unitary groups linked to self-adjoint extensions of differential operators, shedding light on geometric conditions for spectral sets related to the Fuglede conjecture.

Contribution

It provides a new formula for unitary groups associated with differential operators on unions of intervals and explores geometric criteria for spectral sets in relation to the Fuglede conjecture.

Findings

Derived a formula for unitary groups on unions of intervals.

Identified geometric properties of spectral sets in $r$.

Connected spectral set properties to the Fuglede conjecture.

Abstract

In connection to the Fuglede conjecture, and to Fuglede's original work \cite{Fug74}, we study one-parameter unitary groups associated to self-adjoint extensions of the differential operator $Df=\frac1{2\pi i}f'$ on a union of finite intervals. We present a formula for such unitary groups and we use it to discover some geometric properties of such sets in $\br$ which admit orthogonal bases of exponential functions (also called spectral sets).