Dimensional reduction formulae for spectral traces and Casimir energies

This short letter considers the case of acoustic scattering by several obstacles in ${\mathbb{R}}^{d+r}$ for $r,d\ge 1$ of the form $\mathrm{\Omega }\times {\mathbb{R}}^r$, where $\mathrm{\Omega }$ is a smooth bounded domain in ${\mathbb{R}}^d$. 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 $\mathrm{\Omega }\times {\mathbb{R}}^r$ per unit volume in ${\mathbb{R}}^r$.