Equidistribution of phase shifts in obstacle scattering

Abstract

For scattering off a smooth, strictly convex obstacle with positive curvature, we show that the eigenvalues of the scattering matrix – the phase shifts – equidistribute on the unit circle as the frequency at a rate proportional to , under a standard condition on the set of closed orbits of the billiard map in the interior. Indeed, in any sector not containing 1, there are eigenvalues for k large, where cd is a constant depending only on the dimension. Using this result, the two term asymptotic expansion for the counting function of Dirichlet eigenvalues, and a spectral-duality result of Eckmann-Pillet, we then give an alternative proof of the two term asymptotic of the total scattering phase due to Majda-Ralston.