Vacuum energy of Schr\"odinger operators on metric graphs

TL;DR

This paper develops an integral formulation for the vacuum energy of Schrödinger operators on finite metric graphs, linking quantum spectral properties with classical dynamics, and clarifies the conditions under which the Casimir force is well-defined.

Contribution

It introduces an analytic method to compute vacuum energy on metric graphs using zeta functions and classifies vertex matching conditions, addressing ambiguities in the energy calculation.

Findings

Vacuum energy can be expressed via the zeta function derived from secular equations.

The Casimir force on a bond with localized potential is well-defined despite energy ambiguities.

The approach connects quantum spectral data with classical chaotic dynamics.

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.