Year: 2024
Type: article
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.1. Introduction 1019 2. Main theorem and applications 1030 3. Preliminaries 1036 4. Construction of a refined quantum partition 1062 5. Reduction to a fractal uncertainty principle via microlocalization properties 1092 6. Application of the fractal uncertainty principle 1111
Source: Analysis & PDE
Author: Lucas Vacossin
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4395067423