Direct and inverse problems for a two-dimensional metamaterial-modified transmission eigenvalue problem

In this work we study the two dimensional modified transmission eigenvalue problem corresponding to a metamaterial. For the direct problem, we consider a radially symmetric refractive index n(r) and show the existence of infinitely many eigenvalues for each characteristic function through asymptotics of the Bessel functions. Then, we are able to reconstruct the Dirichlet-to-Neumann map for an auxiliary Helmholtz problem with inhomogeneity n(r) by using the knowledge of only one eigenvalue from each characteristic. This provides a uniqueness theorem for the refractive index, by utilizing the Borg-Levinson theory for potentials of the Schrödinger equation in two dimensions. We solve numerically the direct problem, to compute the eigenvalues given the n(r), by constructing a spectral finite element scheme. These values could be used as input for the corresponding inverse problem. In particular, we address the problem of reconstructing radially symmetric refractive indices (both continuous and piecewise continuous) by solving an optimization problem that minimizes the error between the measured and approximated Dirichlet-to-Neumann maps.