The Boundary-Integral Formulation and Multiple-Reflection Expansion for the Vacuum Energy of Quantum Graphs

TL;DR

This paper explores boundary-integral methods for calculating vacuum energy in quantum graphs, connecting classical-path and boundary integral approaches to improve understanding and computation of spectral functions.

Contribution

It demonstrates how known solutions for Kirchhoff quantum graphs can be derived using a boundary-integral formulation, bridging different analytical methods.

Findings

Boundary-integral formulation reproduces known Kirchhoff quantum graph solutions

Establishes connections between classical-path and boundary integral approaches

Provides a foundation for improved spectral function calculations

Abstract

Vacuum energy and other spectral functions of Laplace-type differential operators have been studied approximately by classical-path constructions and more fundamentally by boundary integral equations. As the first step in a program of elucidating the connections between these approaches and improving the resulting calculations, I show here how the known solutions for Kirchhoff quantum graphs emerge in a boundary-integral formulation.