Spectral gap for obstacle scattering in dimension 2

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.