The Birman-Krein formula for differential forms and electromagnetic scattering

TL;DR

This paper extends the Birman-Krein formula to differential forms on Riemannian manifolds with Euclidean ends, including Maxwell scattering in three dimensions, providing a new analytical framework for electromagnetic scattering analysis.

Contribution

It establishes a Birman-Krein formula for the Laplace Beltrami operator on differential forms on manifolds with low-regularity boundaries, linking scattering theory and geometric analysis.

Findings

Proves Birman-Krein formula for co-closed differential forms on manifolds with Euclidean ends.

Specializes the formula to Maxwell scattering in three dimensions.

Handles low regularity boundaries in the scattering setting.

Abstract

We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean near infinity. Allowing for compact boundaries of low regularity we prove a Birman-Krein formula on the space of co-closed differential forms. In the case of dimension three this reduces to a Birman-Krein formula in Maxwell scattering.