We prove the existence of an infinite discrete set of transmission eigenvalues corresponding to the scattering problem for isotropic and anisotropic inhomogeneous media for both the Helmholtz and Maxwell's equations. Our discussion includes the case of the interior transmission problem for an inhomogeneous medium with cavities, i.e. subregions with contrast zero.

The recovery of a spherically-symmetric wave speed v is considered in a bounded spherical region of radius b from the set of the corresponding transmission eigenvalues for which the corresponding eigenfunctions are also spherically symmetric. If the integral of 1/v on the interval [0, b] is less than b, assuming that there exists at least one v corresponding to the data, it is shown that v is uniquely determined by the data consisting of such transmission eigenvalues and their "multiplicities," where the "multiplicity" is defined as the multiplicity of the transmission eigenvalue as a zero of a key quantity.When that integral is equal to b, the unique recovery is obtained when the data contains one additional piece of information. Some similar results are presented for the unique determination of the potential from the transmission eigenvalues with "multiplicities" for a related Schrödinger equation. Mathematics Subject Classification (2010): 34B07 34B24 47E05Short title: Inverse problem for transmission eigenvalues

We consider the inverse problem of determining the spherically symmetric index of refraction n(r) from a knowledge of the corresponding transmission eigenvalues (which can be determined from field pattern of the scattered wave). We also show that for constant index of refraction n(r) = n, the smallest transmission eigenvalue suffices to determine n, complex eigenvalues exist for n sufficiently small and, for homogeneous media of general shape, determine a region in the complex plane where complex eigenvalues must lie.

The present work is concerned with the extension of the factorization method to the inverse elastic scattering problem by penetrable isotropic bodies embedded in an isotropic host environment for time-harmonic plane wave incidence. Although the former method has been successfully employed for the shape reconstruction problem in the field of elastodynamic scattering by rigid bodies or cavities, no corresponding results have been recorded, so far, for the very interesting (both from a theoretical and a practical point of view) case of isotropic elastic inclusions. This paper aims at closing this gap by developing the theoretical framework which is necessary for the application of the factorization method to the inverse transmission problem in elastodynamics. As in the previous works referring to the particular reconstruction method, the main outcome is the construction of a binary criterion which determines whether a given point is inside or outside the scattering obstacle by using only the spectral data of the far-field operator.

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