Relative trace formulas for obstacle scattering with Neumann and transmission boundary conditions

Arne Hofmann; Alexander Strohmaier

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arXiv · 2605.21201

Relative trace formulas for obstacle scattering with Neumann and transmission boundary conditions

Arne Hofmann·Alexander Strohmaier

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Abstract

We consider the case of scattering by several obstacles in $\mathbb{R}^d$ for $d \geq 2$. We establish a relative trace formula for Neumann and transmission boundary conditions analogous to the one obtained in arXiv:2002.07291 for Dirichlet boundary conditions. In the case of $f(x) = x^{1/2}$ the trace has the interpretation of the Casimir energy of the obstacle configuration. In the one-dimensional case, we recover a rigorous version of the Lifshitz formula for the Casimir energy of parallel plates with frequency-independent electric permittivity and magnetic permeability. We thereby strengthen the mathematical foundations of the Casimir effect and demonstrate the flexibility of the rigorous approach established in arXiv:2104.09763 and arXiv:2002.07291.

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Arne Hofmann, Alexander Strohmaier. 2026-05-20. Relative trace formulas for obstacle scattering with Neumann and transmission boundary conditions. https://arxiv.org/abs/2605.21201

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