English

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

Mathematical Physics 2026-05-21 v1 Analysis of PDEs math.MP Spectral Theory

Abstract

We consider the case of scattering by several obstacles in Rd\mathbb{R}^d for d2d \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)=x1/2f(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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Cite

@article{arxiv.2605.21201,
  title  = {Relative trace formulas for obstacle scattering with Neumann and transmission boundary conditions},
  author = {Arne Hofmann and Alexander Strohmaier},
  journal= {arXiv preprint arXiv:2605.21201},
  year   = {2026}
}

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50 pages