Vacuum energy of Schr\"odinger operators on metric graphs
Abstract
We present an integral formulation of the vacuum energy of Schr\"odinger operators on finite metric graphs. Local vertex matching conditions on the graph are classified according to the general scheme of Kostrykin and Schrader. While the vacuum energy of the graph can contain finite ambiguities the Casimir force on a bond with compactly supported potential is well defined. The vacuum energy is determined from the zeta function of the graph Schr\"odinger operator which is derived from an appropriate secular equation via the argument principle. A quantum graph has an associated probabilistic classical dynamics which is generically both ergodic and mixing. The results therefore present an analytic formulation of the vacuum energy of this quasi-one-dimensional quantum system which is classically chaotic.
Keywords
Cite
@article{arxiv.1201.3754,
title = {Vacuum energy of Schr\"odinger operators on metric graphs},
author = {J. M. Harrison and K. Kirsten},
journal= {arXiv preprint arXiv:1201.3754},
year = {2015}
}
Comments
10 pages, to appear in proceedings of QFEXT11 (IJMPCS, World Scientific)