In this work, we consider the inverse transmission eigenvalue problem to determine the refractive index from transmission eigenvalues. We adopt a weak formulation of the problem and provide a Galerkin scheme in to compute transmission eigenvalues. Using a proper operator representation of the problem, we show convergence of the method. Next, we define the inverse transmission problem and show that numerically the problem can be considered as a discrete inverse quadratic eigenvalue problem. First, we investigate the case of a spherically symmetric piecewise constant refractive index and confirm our results with analytic computations. Then, we show that a relative small number of eigenvalues are sufficient for simple cases of a few layers by just minimizing the total error between measured and computed eigenvalues to reconstruct the refractive index. Finally, we propose a computational method based on a Newton-type algorithm for reconstructions of a general piecewise constant refractive index for any domain from transmission eigenvalues. The algorithm can be performed without having knowledge of the exact position of the eigenvalues in the spectrum.

We consider the inverse spectral problem of determining a spherically symmetric and discontinuous refractive index n ( r ) from interior transmission eigenvalues. Using Liouville’s transform, we investigate the asymptotic properties of the solution of an auxiliary initial value problem for large wave numbers and the asymptotic behaviour of the characteristic determinants derived from the eigenfunction expansions. Next, we assume that we know all transmission eigenvalues with spherically symmetric eigenfunctions and prove under some conditions that the transformed discontinuity of the refractive index can be determined. Finally we prove that the knowledge of all transmission eigenvalues including multiplicities uniquely determines n ( r ), under the assumption that n (0) is known and either n ( r ) > 1 or 0 < n ( r ) < 1 by using a moment type result and applying Müntz’s theorem.

In the present work we study the modified transmission eigenvalue problem for the case of the spherically symmetric refractive index. This problem occurs as a modification of the classic transmission eigenvalue problem for a fixed wavenumber, by introducing a new spectral parameter. We prove the existence of infinite and discrete eigenvalues using asymptotic expressions and monotonicity properties of the characteristic functions. Next, we define the inverse spectral problem of determining the refractive index from the knowledge of modified transmission eigenvalues. We show that one eigenvalue corresponding to each characteristic function is sufficient to uniquely determine the fraction of the boundary Cauchy data. The knowledge of these data, in combination with a Dirichlet-to-Neumann mapping property allows us to establish uniqueness for the inverse problem.