The solution of the "constant term problem" and the zeta-regularized determinant for quantum graphs

Abstract

We generalize the "Fulling-Kuchment-Wilson formula" involving the "constant term" of the general trace formula for quantum graphs. Furthermore, we derive the zeta-regularized determinant, calculate the corresponding vacuum energy for the quantized Klein-Gordon field on a metric graph and obtain an explicit expression for the corresponding Casimir force.