Zeta Functions of the Dirac Operator on Quantum Graphs

TL;DR

This paper develops spectral zeta functions for the Dirac operator on quantum graphs, extending contour integral methods used for Laplace and Schrödinger operators to a broader class of operators on metric graphs.

Contribution

It introduces a construction of spectral zeta functions for the Dirac operator on finite metric graphs, including a regularized spectral determinant, using contour integral techniques.

Findings

Spectral zeta functions are formulated for the Dirac operator on various graphs.

The regularized spectral determinant is obtained as the derivative of the zeta function.

Method extends existing techniques for Laplace and Schrödinger operators to Dirac operators.

Abstract

We construct spectral zeta functions for the Dirac operator on metric graphs. We start with the case of a rose graph, a graph with a single vertex where every edge is a loop. The technique is then developed to cover any finite graph with general energy independent matching conditions at the vertices. The regularized spectral determinant of the Dirac operator is also obtained as the derivative of the zeta function at a special value. In each case the zeta function is formulated using a contour integral method, which extends results obtained for Laplace and Schrodinger operators on graphs.