An improved fractal Weyl upper bound in obstacle scattering

Abstract 

In this talk, we will be interested in the problem of scattering by strictly convex obstacles in 2D and we will give estimates for the number of resonances in regions $\{ \re \lambda \in [r, r+1], \im \lambda \geq - \gamma \}$ in the limit $r \to + \infty$. It is now well known in open quantum chaos that the dimension of the trapped set appears when counting resonances in such boxes : the number of resonances is $O(r^\delta)$ where \delta is related with the dimension of the trapped set. This result is often called a Fractal Weyl upper bound. Nevertheless, it is expected that this bound is not optimal when $\gamma$ is below some threshold and Improved Fractal Weyl upper bounds have emerged in different contexts of open quantum chaos (convex co-compact hyperbolic surfaces, open baker's maps). In this presentation, we will explain a version of such a result in obsctacle scattering. We will explain the main ideas of the proof, which relies on a subtle interaction between the propagation of coherent states and the use of escape functions.

About the speaker 

I am currently a PHD Student under the supervision of Stéphane Nonnenmacher at Laboratoire de Mathématiques d'Orsay (Université Paris-Saclay), in the team AN-EDP. I am also AGPR at l'Ecole Normale Supérieure (Paris).

I am intereted in the study of resonances in open hyperbolic systems. In particular, I study the problem of scattering by obstacles, where I am interested in the distribution of the resonances of the Laplace operator outside the obstacles, which is known to have consequences on the wave equation outside the obstacles.

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https://www.math.ens.psl.eu/~vacossin/