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Dimensional reduction formulae for spectral traces and Casimir energies

Alexander Strohmaier

Abstract

This short letter considers the case of acoustic scattering by several obstacles in $\mathbb{R}^{d+r}$ for $r,d \geq 1$ of the form $Ω\times \mathbb{R}^r$, where $Ω$ 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 $Ω\times \mathbb{R}^r$ per unit volume in $\mathbb{R}^r$.

Dimensional reduction formulae for spectral traces and Casimir energies

Abstract

This short letter considers the case of acoustic scattering by several obstacles in for of the form , where is a smooth bounded domain in . 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 per unit volume in .
Paper Structure (4 sections, 1 theorem, 30 equations, 1 figure)

This paper contains 4 sections, 1 theorem, 30 equations, 1 figure.

Table of Contents

  1. Introduction: Casimir energy as a relative trace
  2. Neumann trace and dimensional reduction
  3. Trace-norm estimate
  4. Proof of the main theorem

Key Result

Theorem 1.2

The operator $\tilde{D}_s$ is trace-class with respect to the von-Neumann trace $\widetilde{\mathrm{tr}}$ for any $m\geq 0, s>0$. The corresponding trace equals

Figures (1)

  • Figure 1: The contour $\tilde{\Gamma}$. The shaded region is mapped to a sector containing the spectrum of $\Delta$ under the map $z \mapsto z^2$.

Theorems & Definitions (2)

  • Definition 1.1
  • Theorem 1.2