The Dirac Operator on Regular Metric Trees

TL;DR

This paper analyzes the spectral properties of the Dirac operator on regular metric trees, leveraging their symmetries to decompose the space and facilitate detailed spectral analysis.

Contribution

It introduces a method to decompose the Dirac operator on regular metric trees using their symmetries, enabling comprehensive spectral analysis.

Findings

Spectral properties depend on the geometry of the tree

Symmetry-based decomposition simplifies analysis

Detailed spectral characterization achieved

Abstract

A metric tree is a tree whose edges are viewed as line segments of positive length. The Dirac operator on such tree is the operator which operates on each edge, complemented by the matching conditions at the vertices which were given by Bolte and Harrison \cite{BolteHarrison2003}. The spectrum of Dirac operator can be quite different, reflecting geometry of the tree. We discuss a special case of trees, namely the so-called regular trees. They possess a rich group of symmetries. This allows one to construct an orthogonal decomposition of the space $L^2(\Gamma)$ which reduces the Dirac. Based upon this decomposition, a detailed spectral analysis of Dirac operator on the regular metric trees is possible.