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 \(\Omega \times \mathbb {R}^r\), where \(\Omega \) 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 \(\Omega \times \mathbb {R}^r\) per unit volume in \(\mathbb {R}^r\).