A perturbation problem for transmission eigenvalues

In this paper, we consider a perturbation problem for real transmission eigenvalues. Real transmission eigenvalues are of particular interest in inverse scattering theory. They can be determined from scattering data and are related to injectivity of the related scattering operators. The goal of this paper is to provide examples of existence of real transmission eigenvalues for inhomogeneities whose refractive index does not satisfy the assumptions for which the (non-self-adjoint) transmission eigenvalue problem is understood. Such “irregular media” are obtained as perturbations of an inhomogeneity for which the existence of real transmission eigenvalues is known. Our perturbation approach uses an application of a version of the implicit function theorem to an appropriate function in the vicinity of an unperturbed real transmission eigenvalue. Several examples of interesting spherical perturbations of spherically symmetric media are included. Partial results are obtained for general media based on our perturbation approach.