The heat kernel on the diagonal for a compact metric graph
Abstract
We analyze the heat kernel associated to the Laplacian on a compact metric graph, with standard Kirchoff-Neumann vertex conditions. An explicit formula for the heat kernel as a sum over loops, developed by Roth and Kostrykin, Potthoff, and Schrader, allows for a straightforward analysis of small-time asymptotics. We show that the restriction of the heat kernel to the diagonal satisfies a modified version of the heat equation. This observation leads to an "edge" heat trace formula, expressing the a sum over eigenfunction amplitudes on a single edge as a sum over closed loops containing that edge. The proof of this formula relies on a modified heat equation satisfied by the diagonal restriction of the heat kernel. Further study of this equation leads to explicit formulas for completely symmetric graphs.
Cite
@article{arxiv.2204.06619,
title = {The heat kernel on the diagonal for a compact metric graph},
author = {David Borthwick and Kenny Jones and Evans M. Harrell},
journal= {arXiv preprint arXiv:2204.06619},
year = {2023}
}
Comments
17 pages, 7 figures. Revision includes minor corrections and additional references