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 $c_d\left|S\right|\text{Vol}(\partial \Omega )k^{d-1}+o(k^{d-1})$ 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.

Keywords:

• Equidistribution
• Laplacian
• obstacle scattering
• phase shifts

MATHEMATICS SUBJECT CLASSIFICATION:

• 35P20
• 35P25

Additional information

Funding

J.G.R. acknowledges the support of the Australian Research Council through Discovery Grant DP180100589. M.I. was funded by the LabEx IRMIA, and partially supported by the Agence Nationale de la Recherche project GeRaSic (ANR-13-BS01-0007-01). Both authors wish to thank the Australian Mathematical Sciences Institute and the Mathematical Sciences Institute at the Australian National University for their partial funding of the workshop “Microlocal Analysis and its Applications in Spectral Theory, Dynamical Systems, Inverse Problems, and PDE” at which part this project was completed.