[Dispersion for the wave equation in the exterior of the torus]{Dispersion for the wave equation in the exterior of the torus in three dimensions}
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916748389777408 |
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| author | Quirchmayr, Ronald Waters, Alden |
| author_facet | Quirchmayr, Ronald Waters, Alden |
| contents | We prove dispersive estimates for the wave equation in the exterior of a torus. Because no separation of variables into a basis of eigenfunctions and eigenvalues exists for the time harmonic problem, we introduce a related approximate operator for the Dirichlet Laplacian in the exterior of a torus. The approximate operator coincides with the Schrödinger operator with a Pöschl-Teller potential and agrees with the Dirichlet Laplacian to leading order. The operator here which we develop is related to the so-called Mehler-Fock kernel. Using the known solution to the eigenvector and eigenvalue problem of Pöschl-Teller, a high-frequency analysis of the approximate operator for the wave equation can be made accurately. The operator for this problem gives a close approximation to the $L^1\rightarrow L^{\infty}$ dispersive estimate at a suitable small distance from the torus for the corresponding exterior wave operator with Dirichlet Laplacian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15264 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | [Dispersion for the wave equation in the exterior of the torus]{Dispersion for the wave equation in the exterior of the torus in three dimensions} Quirchmayr, Ronald Waters, Alden Analysis of PDEs Spectral Theory We prove dispersive estimates for the wave equation in the exterior of a torus. Because no separation of variables into a basis of eigenfunctions and eigenvalues exists for the time harmonic problem, we introduce a related approximate operator for the Dirichlet Laplacian in the exterior of a torus. The approximate operator coincides with the Schrödinger operator with a Pöschl-Teller potential and agrees with the Dirichlet Laplacian to leading order. The operator here which we develop is related to the so-called Mehler-Fock kernel. Using the known solution to the eigenvector and eigenvalue problem of Pöschl-Teller, a high-frequency analysis of the approximate operator for the wave equation can be made accurately. The operator for this problem gives a close approximation to the $L^1\rightarrow L^{\infty}$ dispersive estimate at a suitable small distance from the torus for the corresponding exterior wave operator with Dirichlet Laplacian. |
| title | [Dispersion for the wave equation in the exterior of the torus]{Dispersion for the wave equation in the exterior of the torus in three dimensions} |
| topic | Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2505.15264 |