# Equidistribution of Phase Shifts in Obstacle Scattering

## TL;DR

This paper proves that phase shifts in obstacle scattering become uniformly distributed on the unit circle at high frequencies for convex obstacles, providing new insights into spectral asymptotics and scattering theory.

## Contribution

It establishes the equidistribution of scattering matrix eigenvalues for convex obstacles and offers an alternative proof of the total scattering phase asymptotics.

## Key findings

- Eigenvalues of the scattering matrix equidistribute on the unit circle as frequency increases.
- Number of eigenvalues in sectors scales with $k^{d-1}$ and the sector size.
- Provides an alternative proof of the two-term asymptotic expansion of the total scattering phase.

## 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.