Dimensional reduction formulae for spectral traces and Casimir energies

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