Heat kernels on metric graphs and a trace formula

TL;DR

This paper derives a heat kernel representation and trace formula for Laplace operators on metric graphs with spectral parameter-independent scattering matrices, with applications to inverse spectral and scattering problems.

Contribution

It introduces a novel heat kernel sum-over-walks representation and a trace formula for a specific class of Laplace operators on metric graphs.

Findings

Heat kernel expressed as sum over walks between edges

Trace formula established for heat semigroups

Applications to inverse spectral and scattering problems

Abstract

We study heat semigroups generated by self-adjoint Laplace operators on metric graphs characterized by the property that the local scattering matrices associated with each vertex of the graph are independent from the spectral parameter. For such operators we prove a representation for the heat kernel as a sum over all walks with given initial and terminal edges. Using this representation a trace formula for heat semigroups is proven. Applications of the trace formula to inverse spectral and scattering problems are also discussed.