2023/01/26 by Lucas Vacossin, Vacossin, Lucas · 1 citation
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2301.11077
openalex publication_date 2023/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we are interested in the problem of scattering by strictly convex obstacles in the plane. We provide an upper bound for the number N(r,γ) of resonances in the box \r ≤ \Re(λ) ≤ r + 1; \Im(λ) ≥ - γ\. It was proved in a work of S. Nonnenmacher, J. Sjöstrand and M. Zworski (2014) that N (r,γ) = Oγ(rdH) where 2dH + 1 is the Hausdorff dimension of the trapped set of the billiard flow. In this article, we provide an improved upper bound in the band 0 ≤ γ< γcl /2, where γcl is the classical decay rate of the flow. This improved Weyl upper bound is in the spirit of the ones of F. Naud (2012) and S. Dyatlov (2019) in the case of convex co-compact surfaces, and of S. Dyatlov and L. Jin (2017) in the case of open quantum baker's maps.