Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 022, 23 pages      arXiv:2307.13324      https://doi.org/10.3842/SIGMA.2024.022

Boundary Value Problems for Dirac Operators on Graphs

Alberto Richtsfeld
Institut für Mathematik, Universität Potsdam, D-14476, Potsdam, Germany

Received August 21, 2023, in final form February 28, 2024; Published online March 19, 2024

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.

Key words: metric graphs; Dirac operator; boundary value problems.

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