Scattering Resonances for Convex Obstacles

In the setting of obstacle scattering in Euclidean spaces, the poles of meromorphic continuation of the resolvent of the Laplacian on the exterior region are called the resonances or scattering poles. Each resonance corresponds to a resonant wave. The real part of a resonance corresponds to the frequency of the wave, while the imaginary part corresponds to the decay rate of the wave. Consequently understanding the distribution of the resonances is important in understanding the long time behavior of the solution to wave equations in the exterior domain. We study the distribution of resonances in the case of a strictly convex obstacle with smooth boundary. In particular, under general boundary conditions, we prove the existence of the cubic resonance free regions near the real axis. Moreover, if the obstacle is close to a sphere, in the sense that it satisfies certain pinched curvature conditions, we prove that the resonances close to the real axis are separated into cubic bands and in each band, the counting function of resonances satisfies a Weyl law. We also generalize these results to totally convex obstacles in more general asymptotic Euclidean metrics