Dimensional reduction formulae for spectral traces and Casimir energies

This short letter considers the case of acoustic scattering by several obstacles in ℝ^d+r for r,d ≥ 1 of the form Ω×ℝ^r, where Ω is a smooth bounded domain in ℝ^d. As a main result a von-Neumann-trace formula for the relative trace is obtained in this setting. As a special case we obtain a dimensional reduction formula for the Casimir energy for the massive and massless scalar fields in this configuration Ω×ℝ^r per unit volume in ℝ^r.