The momentum operator on a union of intervals and the Fuglede conjecture

TL;DR

This paper explores the relationship between the spectral and tiling properties of sets in one dimension, using the momentum operator on unions of intervals to analyze the Fuglede conjecture.

Contribution

It introduces a framework connecting geometric configurations with spectral properties via self-adjoint extensions of the momentum operator in one dimension.

Findings

Connections established between spectrum and tiling for unions of intervals

Characterization of self-adjoint extensions of the momentum operator in this context

Insights into the Fuglede conjecture in one-dimensional settings

Abstract

The purpose of the present paper is to place a number of geometric (and hands-on) configurations relating to spectrum and geometry inside a general framework for the {\it Fuglede conjecture}. Note that in its general form, the Fuglede conjecture concerns general Borel sets $\Omega$ in a fixed number of dimensions $d$ such that $\Omega$ has finite positive Lebesgue measure. The conjecture proposes a correspondence between two properties for $\Omega$, one takes the form of spectrum, while the other refers to a translation-tiling property. We focus here on the case of dimension one, and the connections between the Fuglede conjecture and properties of the self-adjoint extensions of the momentum operator $\frac{1}{2\pi i}\frac{d}{dx}$, realized in $L^2$ of a union of intervals.