On the eigenvalues of the volume integral operator of electromagnetic scattering

The volume integral equation of electromagnetic scattering can be used to c ompute the scattering by inhomogeneous or anisotropic scatterers. In this p aper we compute the spectrum of the scattering integral operator for a sphe re and the eigenvalues of the coefficient matrices that arise from the disc retization of the integral equation. For the case of a spherical scatterer, the eigenvalues lie mostly on a line in the complex plane, with some eigen values lying below the line. We show how the spectrum of the integral opera tor can be related to the well-posedness of a modi ed scattering problem. T he eigenvalues lying below the line segment arise from resonances in the an alytical 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 eigenv alues of the coefficient matrix determine the rate of convergence of iterat ive solvers. We show that when the scatterer is a small sphere, the converg ence rate for the nonnormal coefficient matrices can be estimated but this estimate is no longer valid for large spheres.