Dimensional reduction formulae for spectral traces and Casimir energies

Abstract

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