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Spectral gap for obstacle scattering in dimension 2

Lucas Vacossin

Vol. 17 (2024), No. 3, 1019–1126
Abstract

We study the problem of scattering by several strictly convex obstacles, with smooth boundary and satisfying a noneclipse 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 by Nonnenmacher et al. (Ann. of Math. (2) 179:1 (2014), 179–251). In fact, we obtain a spectral gap for this type of object, which also has applications in potential scattering. The second main ingredient of this article is a fractal uncertainty principle. We adapt the techniques of Dyatlov et al. (J. Amer. Math. Soc. 35:2 (2022), 361–465) to apply this fractal uncertainty principle in our context.

Keywords
scattering resonances, spectral gap, fractal uncertainty principle
Mathematical Subject Classification
Primary: 35P05, 35P25, 35Q40, 35S30, 35J05, 35J10, 37D20
Milestones
Received: 13 January 2022
Revised: 10 June 2022
Accepted: 11 August 2022
Published: 24 April 2024
Authors
Lucas Vacossin
Laboratoire de Mathématiques d’Orsay
Université Paris-Saclay
UMR 8628 du CNRS
Orsay
France

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