Dimensional reduction formulae for spectral traces and Casimir energies

This short letter considers the case of acoustic scattering by several obstacles in Rᵈ⁺ʳ R d + r for r,d ≥ 1 r , d ≥ 1 of the form Ω × Rʳ Ω × R r , where Ω Ω is a smooth bounded domain in Rᵈ 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 × Rʳ Ω × R r per unit volume in Rʳ R r .