The trace formula for quantum graphs with general self adjoint boundary conditions

TL;DR

This paper develops trace formulas for quantum graphs with general self-adjoint boundary conditions, linking spectra to periodic orbits and analyzing heat kernel asymptotics.

Contribution

It introduces new trace formulas for quantum graphs with arbitrary self-adjoint boundary conditions, including convergence conditions and heat kernel asymptotics.

Findings

Derived trace formulas with different convergence properties

Established heat kernel small-t asymptotics

Linked spectral data to periodic orbit sums

Abstract

We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbits on the graph. This includes trace formulae with, respectively, absolutely and conditionally convergent periodic orbit sums; the convergence depending on properties of the test functions used. We also prove a trace formula for the heat kernel and provide small-$t$ asymptotics for the trace of the heat kernel.