Equidistribution of phase shifts in trapped scattering

TL;DR

This paper proves that the eigenvalues of the scattering matrix become uniformly distributed under certain classical dynamical conditions, extending previous results to cases with trapped trajectories.

Contribution

It extends previous equidistribution results for scattering matrix eigenvalues to scenarios with trapped classical trajectories, under zero Liouville measure conditions.

Findings

Eigenvalues of the scattering matrix are equidistributed under specified conditions.

The result generalizes prior work by including trapped sets with zero Liouville measure.

The proof relies on classical dynamical assumptions and microlocal analysis.

Abstract

We prove an equidistribution result for the eigenvalues of the scattering matrix associated to an operator of the form $-h^2\Delta + V-1$, where $V\in C_c^\infty(\mathbb{R}^d)$ is a compactly supported potential, under the assumption that the incoming and outgoing sets of the classical dynamics have zero Liouville measure. This extends a recent result of Gell-Redman, Hassell and Zelditch, where the authors proved equidistribution of the eigenvalues of the scattering matrix under the assumption that the trapped set is empty.