Boundary Value Problems for Dirac Operators on Graphs

TL;DR

This paper extends index theory for manifolds with boundary to Dirac operators on metric graphs, providing a concise proof of their index, analyzing self-adjoint extensions, and linking boundary conditions to graph topology.

Contribution

It introduces a novel application of index theory to Dirac operators on graphs and explores boundary conditions that reveal topological information.

Findings

Short proof for the index of Dirac operators on graphs

Characterization of self-adjoint extensions and spectrum

Boundary conditions encode graph topology

Abstract

We carry the index theory for manifolds with boundary of B\"ar and Ballmann over to first order differential operators on metric graphs. This approach results in a short proof for the index of such operators. Then the self-adjoint extensions and the spectrum of the Dirac operator on the complex line bundle are studied. We also introduce two types of boundary conditions for the Dirac operator, whose spectrum encodes information of the underlying topology of the graph.