Momentum operators on graphs

TL;DR

This paper explores the construction and analysis of momentum operators on oriented metric graphs, establishing conditions for their existence, examining their spectral properties, and highlighting the failure of the unique continuation principle.

Contribution

It introduces a method to define momentum operators on graphs with even degrees, analyzes their spectra, and discusses the implications for quantum graph theory.

Findings

Balanced orientation is necessary for defining momentum operators.

Momentum operators can be constructed on graphs with even vertex degrees.

The spectrum of these operators and the failure of unique continuation are characterized.

Abstract

We discuss ways in which momentum operators can be introduced on an oriented metric graph. A necessary condition appears to the balanced property, or a matching between the numbers of incoming and outgoing edges; we show that a graph without an orientation, locally finite and at most countably infinite, can made balanced oriented \emph{iff} the degree of each vertex is even. On such graphs we construct families of momentum operators; we analyze their spectra and associated unitary groups. We also show that the unique continuation principle does not hold here.