On the Eigenvalues of the Volume Integral Operator of Electromagnetic Scattering

The volume integral equation of electromagnetic scattering can be used to compute the scattering by inhomogeneous or anisotropic scatterers. In this paper we compute the spectrum of the scattering integral operator for a sphere and the eigenvalues of the coefficient matrices that arise from the discretization of the integral equation. For the case of a spherical scatterer, the eigenvalues lie mostly on a line in the complex plane, with some eigenvalues lying below the line. We show how the spectrum of the integral operator can be related to the well-posedness of a modified scattering problem. The eigenvalues lying below the line segment arise from resonances in the analytical series solution of scattering by a sphere. The eigenvalues on the line are due to the branch cut of the square root in the definition of the refractive index. We try to use this information to predict the performance of iterative methods. For a normal matrix the initial guess and the eigenvalues of the coefficient matrix determine the rate of convergence of iterative solvers. We show that when the scatterer is a small sphere, the convergence rate for the nonnormal coefficient matrices can be estimated but this estimate is no longer valid for large spheres.