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Abstract In this work, we construct the Born and inverse Born approximation and series to recover two function-valued coefficients in the Helmholtz equation for inverse scattering problems from the scattering data at two different wave-numbers. An analysis of the convergence and approximation error of the proposed regularized inverse Born series is provided. The results show that the proposed series converges when the inverse Born approximations of the perturbations are sufficiently small. The preliminary numerical results show the capability of the proposed regularized inverse Born approximation and series for recovering the isotropic inhomogeneous media. 

This short note was motivated by our eorts to investigate whether there exists a half plane free of transmission eigenvalues for Maxwell's equations. This question is related to solvability of the time domain interior transmission problem which plays a fundamental role in the justi cation of linear sampling and factorization methods with time dependent data. Our original goal was to adapt semiclassical analysis techniques developed in [21, 23] to prove that for some combination of electromagnetic parameters, the transmission eigenvalues lie in a strip around the real axis. Unfortunately we failed. To try to understand why, we looked at the particular example of spherically symmetric media, which provided us with some insight on why we couldn't prove the above result. Hence this paper reports our ndings on the location of all transmission eigenvalues and the existence of complex transmission eigenvalues for Maxwell's equations for spherically strati ed media. We hope that these results can provide reasonable conjectures for general electromagnetic media.