On the modified transmission eigenvalue problem with an artificial metamaterial background

The modified transmission eigenvalue problem arises in inverse scattering theory for inhomogeneous media, by embedding the relevant medium into an other, artificially introduced inhomogeneous medium. In the present work, we examine the case where the artificial medium is characterised as a metamaterial, i.e. having a negative valued refractive index. Our aim is to construct an appropriate spectral Galerkin method to compute the modified transmission eigenvalues, with potential use to the inverse spectral problem as well. We show that the modified transmission eigenvalue problem corresponds to a compact and self-adjoint operator for which the eigenfuction system is not complete in the solution space. By introducing an auxiliary Dirichlet–Neumann eigenvalue problem, we construct an eigenfuction system which has the desired completeness property. We use this complete system to define the Galerkin scheme and by applying some results for compact and positive operator eigenvalue problems, we prove the convergence of our method. We present some numerical examples which validate the eigenvalues approximation. Finally, we pose the corresponding inverse spectral problem and show that the largest eigenvalue can determine an unknown constant refractive index.