Relative trace formulas for obstacle scattering with Neumann and transmission boundary conditions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914583201972224 |
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| author | Hofmann, Arne Strohmaier, Alexander |
| author_facet | Hofmann, Arne Strohmaier, Alexander |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_21201 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Relative trace formulas for obstacle scattering with Neumann and transmission boundary conditions Hofmann, Arne Strohmaier, Alexander Mathematical Physics Analysis of PDEs Spectral Theory 35P25, 81T55, 35J25, 35J05 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. |
| title | Relative trace formulas for obstacle scattering with Neumann and transmission boundary conditions |
| topic | Mathematical Physics Analysis of PDEs Spectral Theory 35P25, 81T55, 35J25, 35J05 |
| url | https://arxiv.org/abs/2605.21201 |