Commuting self-adjoint extensions of the partial differential operators on disconnected sets

TL;DR

The paper extends the Segal–Fuglede framework to arbitrary open sets in $\mathbb{R}^d$, including disconnected and unbounded domains, by developing a unitary-group/ spectral-measure approach that characterizes when commuting self-adjoint extensions $H_j$ of $D_j$ exist. Central to the method is a generalized Fourier transform $\mathcal{F}_C$ mapping $L^2(\Omega)$ onto a direct-integral space $L^2(\mu,m)$, yielding $H_j=\mathcal{F}_C^{-1}M_{\lambda_j}\mathcal{F}_C$ and a joint spectrum tied to $\mu$, with multiplicity $m(\lambda)$. The paper then relates spectrality to local translation properties of the associated unitary groups, and develops detailed structure results: atomic, separated spectra in the finite-measure/finitely-many-components case; explicit 1D endpoint-transition dynamics via a boundary matrix $B$; and a tiling–pair-measure equivalence for discrete subgroup tilings, recovering Fuglede’s lattice results as a special case. Collectively, these results broaden Fuglede-type dualities between spectral sets and tilings to disconnected and more general domains, providing concrete spectral decompositions and translation representations tied to geometric structure.

Abstract

In connection with the Fuglede conjecture, we study the existence of commuting self-adjoint extensions of the partial differential operators on arbitrary, possibly disconnected domains in $\br^d$, the associated unitary group, the spectral measure and some geometric properties.

Theorems & Definitions (67)

• Theorem 1.1
• Definition 1.2
• Conjecture 1.3
• Definition 1.4
• Theorem 1.5
• Definition 1.6
• Theorem 1.7
• Definition 2.1
• Definition 2.2
• Proposition 2.3