We investigate the existence of commuting self-adjoint extensions of the partial differential operators \( D_j=\frac{1}{2\pi i}\frac{\partial }{\partial x_j}, \quad j=1,\ldots ,d, \) defined initially on \(C_0^\infty (\Omega )\) , where \(\Omega \subset \mathbb {R}^d\) is an arbitrary open set, possibly disconnected and unbounded. This problem originates in a question of Segal and is closely related to Fuglede’s characterization of spectral sets through commuting self-adjoint extensions and orthogonal Fourier bases. While previous results of Fuglede and Pedersen established a complete correspondence for connected domains, the disconnected case presents new phenomena arising from interactions among distinct connected components. We extend the spectral-theoretic framework to arbitrary open sets and obtain a necessary and sufficient condition for the existence of commuting self-adjoint extensions of the operators \(\{D_j\}\) . The characterization is expressed in terms of a Radon measure, a multiplicity function, and a measurable matrix field that encode the spectral decomposition of the resulting commuting family. In contrast with the connected case, multiplicities greater than one may occur, and generalized eigenfunctions involve component-dependent coefficients attached to the exponential functions \(e_\lambda (x)=e^{2\pi i\lambda \cdot x}\) . We further establish a characterization of spectral sets in terms of strongly continuous unitary representations acting as groups of local translations. For arbitrary open sets, we show that the existence of a group of local translations is equivalent to the spectrality of the domain. This extends the classical connection between spectral pairs, Fourier analysis, and translation symmetries beyond the connected setting. For open sets of finite measure with finitely many connected components, we prove that the associated spectral measure is atomic and that the joint spectrum is separated. We obtain an orthonormal basis consisting of piecewise exponential functions whose amplitudes may vary from one connected component to another. In the one-dimensional setting, where \(\Omega \) is a countable union of intervals, we describe the corresponding unitary groups through a boundary transition matrix governing the propagation of translations across interval endpoints. Finally, for sets that tile \(\mathbb {R}^d\) by a discrete subgroup, we characterize the associated pair measures and derive an explicit formula for the corresponding group of local translations, thereby extending a classical result of Fuglede from lattices to arbitrary discrete subgroups.
