This short letter considers the case of acoustic scattering by several obstacles in Rᵈ⁺ʳ for r,d ≥ 1 of the form Ω × Rʳ, where Ω is a smooth bounded domain in Rᵈ. 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 per unit volume in Rʳ.
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Alexander Strohmaier (2024) studied this question.
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