论文标题
对角上的热核，用于紧凑的公制图
The heat kernel on the diagonal for a compact metric graph
论文作者
论文摘要
我们在紧凑的公制图上分析了与标准的Kirchoff-Neumann顶点条件上与Laplacian相关的热内核。由Roth和Kostrykin，Potthoff和Schrader开发的热内核的明确公式，可以简单地分析小渐近学。我们表明，对对角线的热内核的限制满足了加热方程的修改版本。该观察结果导致“边缘”热量痕迹公式，表达在单个边缘上的本征函数上的a总和作为包含该边缘的封闭环上的总和。该公式的证明依赖于通过对角限制热核所满足的改良热方程。对该方程式的进一步研究导致针对完全对称图的明确公式。
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.
