Spectral properties of unions of intervals and groups of local translations

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 $Df=\frac{1}{2\pi i}{f}^{\mathrm{\prime }}$ 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 $\text{br}$ which admit orthogonal bases of exponential functions (also called spectral sets).