Zeta functions of quantum graphs

TL;DR

This paper develops a general method to construct zeta functions for quantum graphs, enabling analysis of spectral properties and expanding existing results in spectral theory and quantum graph research.

Contribution

It introduces a contour integral technique for zeta functions of quantum graphs, extending to general vertex conditions and unifying spectral analysis methods.

Findings

Derived explicit zeta functions for various quantum graphs.

Obtained new formulas for spectral determinants, vacuum energy, and heat kernel coefficients.

Unified approach broadens applicability of spectral analysis in quantum graph theory.

Abstract

In this article we construct zeta functions of quantum graphs using a contour integral technique based on the argument principle. We start by considering the special case of the star graph with Neumann matching conditions at the center of the star. We then extend the technique to allow any matching conditions at the center for which the Laplace operator is self-adjoint and finally obtain an expression for the zeta function of any graph with general vertex matching conditions. In the process it is convenient to work with new forms for the secular equation of a quantum graph that extend the well known secular equation of the Neumann star graph. In the second half of the article we apply the zeta function to obtain new results for the spectral determinant, vacuum energy and heat kernel coefficients of quantum graphs. These have all been topics of current research in their own right and in each case this unified approach significantly expands results in the literature.