Momentum Operators in Two Intervals: Spectra and Phase Transition

TL;DR

This paper investigates the spectral properties of the momentum operator on two disjoint intervals, providing a comprehensive classification of its selfadjoint extensions and revealing new spectral pairs linked to geometric configurations.

Contribution

It offers a complete characterization of selfadjoint extensions of the momentum operator on two intervals and introduces a new family of spectral pairs connecting spectrum and geometry.

Findings

Complete classification of selfadjoint extensions

Introduction of new spectral pairs

Link between spectrum and geometric configuration

Abstract

We study the momentum operator defined on the disjoint union of two intervals. Even in one dimension, the question of two non-empty open and non-overlapping intervals has not been worked out in a way that extends the cases of a single interval and gives a list of the selfadjoint extensions. Starting with zero boundary conditions at the four endpoints, we characterize the selfadjoint extensions and undertake a systematic and complete study of the spectral theory of the selfadjoint extensions. In an application of our extension theory to harmonic analysis, we offer a new family of spectral pairs. Compared to earlier studies, it yields a more direct link between spectrum and geometry.