Spectral gap for obstacle scattering in dimension 2
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866917653013069824 |
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| author | Vacossin, Lucas |
| author_facet | Vacossin, Lucas |
| contents | In this paper, we study the problem of scattering by several strictly convex obstacles, with smooth boundary and satisfying a non eclipse condition. We show, in dimension 2 only, the existence of a spectral gap for the meromorphic continuation of the Laplace operator outside the obstacles. The proof of this result relies on a reduction to an open hyperbolic quantum map, achieved in [arXiv:1105.3128]. In fact, we obtain a spectral gap for this type of objects, which also has applications in potential scattering. The second main ingredient of this article is a fractal uncertainty principle. We adapt the techniques of [arXiv:1906.08923] to apply this fractal uncertainty principle in our context. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_08259 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Spectral gap for obstacle scattering in dimension 2 Vacossin, Lucas Spectral Theory Mathematical Physics Analysis of PDEs 35P25 (Primary) 35Qxx, 35S30 (Secondary) In this paper, we study the problem of scattering by several strictly convex obstacles, with smooth boundary and satisfying a non eclipse condition. We show, in dimension 2 only, the existence of a spectral gap for the meromorphic continuation of the Laplace operator outside the obstacles. The proof of this result relies on a reduction to an open hyperbolic quantum map, achieved in [arXiv:1105.3128]. In fact, we obtain a spectral gap for this type of objects, which also has applications in potential scattering. The second main ingredient of this article is a fractal uncertainty principle. We adapt the techniques of [arXiv:1906.08923] to apply this fractal uncertainty principle in our context. |
| title | Spectral gap for obstacle scattering in dimension 2 |
| topic | Spectral Theory Mathematical Physics Analysis of PDEs 35P25 (Primary) 35Qxx, 35S30 (Secondary) |
| url | https://arxiv.org/abs/2201.08259 |