A spin integral equation for electromagnetic and acoustic scattering

TL;DR

This paper introduces a new boundary integral equation method for electromagnetic and acoustic scattering problems that avoids interior resonance issues and works efficiently for different types of wave scattering in three dimensions.

Contribution

The paper proposes a novel integral equation formulation applicable to electromagnetic and acoustic scattering, avoiding breakdown at resonances and using a spin representation for fields.

Findings

The new integral equation is free from interior resonance breakdown.

It applies simultaneously to electromagnetic and acoustic scattering problems.

The operator depends analytically on the wave number and is bounded on natural function spaces.

Abstract

We present a new integral equation for solving the Maxwell scattering problem against a perfect conductor. The very same algorithm also applies to sound-soft as well as sound-hard Helmholtz scattering, and in fact the latter two can be solved in parallel in three dimensions. Our integral equation does not break down at interior spurious resonances, and uses spaces of functions without any algebraic or differential constraints. The operator to invert at the boundary involves a singular integral operator closely related to the three dimensional Cauchy singular integral, and is bounded on natural function spaces and depend analytically on the wave number. Our operators act on functions with pairs of complex two by two matrices as values, using a spin representation of the fields.