The operator equations of Lippmann–Schwinger type for acoustic and electromagnetic scattering problems in L²

Abstract

This article is concerned with the scattering of acoustic and electromagnetic time harmonic plane waves by an inhomogeneous medium. These problems can be translated into volume integral equations of the second kind – the most prominent example is the Lippmann–Schwinger integral equation. In this work, we study a particular class of scattering problems where the integral operator in the corresponding operator equation of Lippmann–Schwinger type fails to be compact. Such integral equations typically arise if the modelling of the inhomogeneous medium necessitates space-dependent coefficients in the highest order terms of the underlying partial differential equation. The two examples treated here are acoustic scattering from a medium with a space-dependent material density and electromagnetic medium scattering where both the electric permittivity and the magnetic permeability vary. In these cases, Riesz theory is not applicable for the solution of the arising integral equations of Lippmann–Schwinger type. Therefore, we show that positivity assumptions on the relative material parameters allow to prove positivity of the arising volume potentials in tailor-made weighted spaces of square integrable functions. This result merely holds for imaginary wavenumber and we exploit a compactness argument to conclude that the arising integral equations are of Fredholm type, even if the integral operators themselves are not compact. Finally, we explain how the solution of the integral equations in L ² affects the notion of a solution of the scattering problem and illustrate why the order of convergence of a Galerkin scheme set up in L ² does not suffer from our L ² setting, compared to schemes in higher order Sobolev spaces.

Keywords:

• scattering theory
• Helmholtz equation
• Maxwell system
• Lippmann–Schwinger
• integral equation