Vacuum Energy in Quantum Graphs
Abstract
We calculate the vacuum energy in quantum graphs. Vacuum energy arose in quantum physics but has an independent mathematical interest as a functional carrying information about the eigenvalue spectrum of a system. A quantum graph is a metric graph with a Hamiltonian applied to it, and recent research in quantum graphs has been directed towards their eigenvalue statistics. Quantum graphs provide an interesting model, intermediate between one-dimensional and higher-dimensional systems, in which we can study aspects of vacuum energy. In order to find the expression for vacuum energy, we use two methods: direct computation with the trace formula and the method of images (i.e. multiple reflection). The latter method also directly gives the vacuum energy density. Both methods are done heuristically for star graphs then rigorously for general graphs. We also discuss some properties of the vacuum energy in quantum graphs including: repulsive Casimir forces, convergence and continuity in bond lengths.
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