Vacuum energy, spectral determinant and heat kernel asymptotics of graph Laplacians with general vertex matching conditions

TL;DR

This paper investigates the spectral properties of Laplace operators on metric graphs, focusing on how vertex matching conditions influence the spectral determinant, vacuum energy, and heat kernel asymptotics, providing insights into quantum chaos models.

Contribution

It presents new results relating vertex matching conditions to spectral determinants, vacuum energy, and heat kernel asymptotics for quantum graph Laplacians.

Findings

Spectral determinant formulas for general graphs

Expressions for vacuum energy in terms of vertex conditions

Asymptotic heat kernel behavior depending on graph structure

Abstract

We consider Laplace operators on metric graphs, networks of one-dimensional line segments (bonds), with matching conditions at the vertices that make the operator self-adjoint. Such quantum graphs provide a simple model of quantum mechanics in a classically chaotic system with multiple scales corresponding to the lengths of the bonds. For graph Laplacians we briefly report results for the spectral determinant, vacuum energy and heat kernel asymptotics of general graphs in terms of the vertex matching conditions.