On the heat content of compact quantum graphs

TL;DR

This work analyzes the heat content for Laplacians on compact finite metric graphs with Dirichlet conditions at a prescribed vertex set ${\mathsf{V}_{\mathrm{D}}}$. The authors derive a Roth-type closed formula expressing the heat content for all $t>0$ as a sum over undirected paths hitting the boundary, yielding ${\mathcal{Q}}_t(\mathcal{G};{\mathsf{V}_{\mathrm{D}}})=|\mathcal{G}| - \frac{2\sqrt{t}}{\sqrt{\pi}}\# {\mathsf{V}_{\mathrm{D}}} + 8\sqrt{t}\sum_{p} \alpha(p) \; H\big(\frac{\ell(p)}{2\sqrt{t}}\big)$, where $H$ is a monotone decaying function and $\alpha(p)$ are scattering coefficients. This formula enables precise small-time asymptotics, a Hadamard-type edge-length sensitivity, and modular graph-surgery principles that yield comparison results for heat content across graphs of different topology. By decomposing the path sum into non-topological and topological contributions, the paper connects spectral data, geometry, and combinatorics of directed paths on graphs, providing a versatile framework for assessing how graph modifications influence heat diffusion. The work also develops domination results for subgraphs and volume-changing operations, underpinning a robust set of tools for analyzing heat content under graph operations.

Abstract

We study the heat content for Laplacians on compact, finite metric graphs with Dirichlet conditions imposed at the "boundary" (i.e., a given set of vertices). We prove a closed formula of combinatorial flavour, as it is expressed as a sum over all closed orbits hitting the boundary. Our approach delivers a small-time asymptotic expansion that delivers information on crucial geometric quantities of the metric graph, much in the spirit of the celebrated corresponding result for manifolds due to Gilkey-van den Berg; but unlike other known formulae based on different methods, ours holds for all times $t>0$ and it displays stronger decay rate in the short time limit. Furthermore, we prove new surgery principles for the heat content and use them to derive comparison principles for the heat content between metric graphs of different topology.

Figures (23)

• Figure 2.1: The graph $\mathcal{G}$ with one Dirichlet condition at $0$ on the left and two Dirichlet conditions at $0$ and $\ell$ on the right (Dirichlet conditions are imposed at the white vertices).
• Figure 2.2: The profiles of $\mathcal{Q}_t([0,\ell];{\mathsf{V}_{\mathrm{D}}})$ for $\ell=3$ and $\# {\mathsf{V}_{\mathrm{D}}}=1$ (left), $\# {\mathsf{V}_{\mathrm{D}}}=2$ (right)...
• Figure 2.3: ...and the profiles of $\frac{\sqrt{\pi}}{2\sqrt{t}}\left(\mathcal{Q}_t([0,\ell];{\mathsf{V}_{\mathrm{D}}})-\ell\right)$, again for $\ell=3$ and $\# {\mathsf{V}_{\mathrm{D}}}=1$ (left), $\# {\mathsf{V}_{\mathrm{D}}}=2$ (right)
• Figure 2.4: The star graph $\mathcal{S}_5$ with $5$ edges and Dirichlet conditions imposed at all white vertices.
• Figure 2.5: The profiles of $\mathcal{Q}_t(\mathcal{S}_n;{\mathsf{V}_{\mathrm{D}}})$ (left) and $\frac{\sqrt{\pi}}{2\sqrt{t}}\left(\mathcal{Q}_t(\mathcal{S}_3;{\mathsf{V}_{\mathrm{D}}})-3\ell\right)$ (right) for $\ell=1$ and $n=3$

Theorems & Definitions (69)

• Theorem A
• Theorem B: Heat content asymptotics
• Theorem C: Caccioppoli-type description of perimeter
• Remark 1
• Remark 2
• Lemma 2.1
• Definition 1
• Remark 3
• Remark 4
• Lemma 2.2