Equidistribution of Phase Shifts in Obstacle Scattering

Abstract

For scattering off a smooth, strictly convex obstacle $\Omega \subset {\mathbb{R}}^d$ with positive curvature, we show that the eigenvalues of the scattering matrix -- the phase shifts -- equidistribute on the unit circle as the frequency $k\to \infty$ at a rate proportional to $k^{d-1}$, under a standard condition on the set of closed orbits of the billiard map in the interior. Indeed, in any sector $S\subset {\mathbb{S}}^1$ not containing $1$, there are $c_d|S|\mathrm{Vol}(\partial \Omega )k^{d-1}+o(k^{d-1})$ eigenvalues for $k$ large, where $c_d$ 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.

14 pages, 2 figures. Final published version. Extensively revised

References in corpus (3)

• The spectral density of the scattering matrix for high energies
• Approximation and equidistribution of phase shifts: spherical symmetry
• Equidistribution of phase shifts in semiclassical potential scattering