Spectral gap for obstacle scattering in dimension 2

Vacossin, Lucas

doi:10.2140/apde.2024.17.1019

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Spectral gap for obstacle scattering in dimension 2

Lucas Vacossin

Vol. 17 (2024), No. 3, 1019–1126

DOI: 10.2140/apde.2024.17.1019

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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Vol. 17, No. 3, 2024