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arXiv · 2204.06619

The heat kernel on the diagonal for a compact metric graph

David Borthwick·Kenny Jones·Evans M. Harrell II
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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.

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BibTeXRIS

David Borthwick, Kenny Jones, Evans M. Harrell II. 2022-04-13. The heat kernel on the diagonal for a compact metric graph. https://doi.org/10.1007/s00023-022-01248-z

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