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

TL;DR

This paper investigates the existence and properties of commuting self-adjoint extensions of differential operators on disconnected domains, with implications for the Fuglede conjecture and spectral theory.

Contribution

It provides new insights into the spectral and geometric properties of these operators on disconnected sets, advancing understanding related to the Fuglede conjecture.

Findings

Existence criteria for commuting self-adjoint extensions

Analysis of the associated unitary groups and spectral measures

Connections between geometric properties and spectral behavior

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.