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Boundary Value Problems for Dirac Operators on Graphs

Alberto Richtsfeld

Abstract

We carry the index theory for manifolds with boundary of Bär 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.

Boundary Value Problems for Dirac Operators on Graphs

Abstract

We carry the index theory for manifolds with boundary of Bär 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.
Paper Structure (9 sections, 29 theorems, 67 equations, 3 figures)

This paper contains 9 sections, 29 theorems, 67 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The index of $\boldsymbol{D_B}$
  4. The scalar Dirac operator
  5. Self-adjoint extensions
  6. The spectrum of the Dirac operator
  7. Boundary conditions
  8. Decompositions of Eulerian directed multigraphs
  9. The directed adjacency matrix

Key Result

Lemma 2.4

For $f,g\in H^1\bigl(\hat{G},\mathbb{C}^r\bigr)$, we have that where $D^\dagger$ is the formal adjoint of $D$ and where $\sigma = (\sigma_e)_{e\in E}$.

Figures (3)

  • Figure 1: The spaces $\mathbb{C}^{\bar{E}}$, $\mathbb{C}^E$ for a graph consisting of a single edge $e$. The continuous function $f$ has boundary values in $\mathbb{C}^{\bar{E}}$.
  • Figure 2: Removal of a vertex.
  • Figure 3: An example of a graph that has two subgraphs $G_1$, $G_{22}$ with no edge going from $G_1$ to $G_{22}$.

Theorems & Definitions (45)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Lemma 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 35 more