The Heat Kernel on the Diagonal for a Compact Metric Graph

We analyze the heat kernel associated with the Laplacian on a compact metric graph, with standard Kirchhoff–Neumann vertex conditions. An explicit formula for the heat kernel as a sum over loops, developed by Roth and Kostrykin–Potthoff–Schrader, allows for a straightforward analysis of small-time a...

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Bibliographic Details
Published inAnnales Henri Poincaré Vol. 24; no. 5; pp. 1661 - 1680
Main Authors Borthwick, David, Harrell II, Evans M., Jones, Kenny
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.05.2023
Springer Nature B.V
Subjects
Classical and Quantum Gravitation
Closed loops
Dynamical Systems and Ergodic Theory
Eigenvectors
Elementary Particles
Formulas (mathematics)
Kernels
Mathematical and Computational Physics
Mathematical Methods in Physics
Original Paper
Physics
Physics and Astronomy
Quantum Field Theory
Quantum Physics
Relativity Theory
Theoretical
Thermodynamics
34B45
81Q35
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Summary:We analyze the heat kernel associated with the Laplacian on a compact metric graph, with standard Kirchhoff–Neumann vertex conditions. An explicit formula for the heat kernel as a sum over loops, developed by Roth and Kostrykin–Potthoff–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 graphs which are symmetric about each vertex.
ISSN:1424-0637
1424-0661
DOI:10.1007/s00023-022-01248-z