二维障碍物散射的谱隙
Spectral gap for obstacle scattering in dimension 2
AI总结：
本文在二维情形下证明了满足非遮蔽条件的多个严格凸光滑障碍物外部Laplace算子亚纯延拓存在谱隙，方法是将问题约化为开双曲量子映射并结合分形不确定性原理。
摘要：
本文研究多个严格凸障碍物的散射问题，这些障碍物具有光滑边界且满足非遮蔽条件。我们仅在二维情形下证明，Laplace算子在障碍物外部亚纯延拓存在谱隙。该结果的证明依赖于将其约化为一个开双曲量子映射，这一约化已在[arXiv:1105.3128]中完成。事实上，我们为这类对象得到了谱隙，该结果在势散射中也有应用。本文的第二个主要成分是一个分形不确定性原理。我们改编[arXiv:1906.08923]中的技术，以便在我们的情境中应用该分形不确定性原理。
英文摘要：
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.
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