Talk by Alberto Richtsfeld
On May 4th, 2023, Alberto Richtsfeld (University of Potsdam, Germany) gave a talk about "Boundary Value Problems for Dirac Operators on Graphs" as part of the online workshop on "Dirac equation between discrete and continuous: new trends and applications" organized by Ginestra Bianconi (Queen Mary University of London) and Delio Mugnolo (FernUniversität in Hagen).
We employ the Bär-Ballmann framework for boundary value problems for first-order differential operators to the Dirac operator on metric graphs. An index formula for general first order operators on metric graphs is easily obtained, generalizing and simplifying the proof of the index formula for Dirac operators given by Post. We continue to study self-adjoint boundary value problems for the Dirac operator on the complex line bundle, finding a formula for the spectrum in the case that all edges of the graph have the same length. We end our discussion by giving two interesting boundary conditions: The first class of boundary conditions is given by permutation matrices which correspond to decompositions of the graph into directed trails. The spectrum of the Dirac operator subject to this condition will contain all the essential information about the decomposition, such as the lengths of the trails appearing. The second boundary condition is given by a new notion of incidence matrix. The spectrum is governed by polynomials which contain valuable information on the cycles present in the graph. For example, these polynomials enable us to directly find the girth of the directed graph.