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Research on transmission eigenvalues has been a very active topic in inverse scattering theory. In this talk, we discuss about the inverse transmission eigenvalue problem for the spherically symmetric refractive index. We present some well-known uniqueness results for the continuous case [1]. Next, we highlight the need to introduce modified problems and demonstrate the corresponding modified transmission eigenvalue problem [2]. A new uniqueness result for the inverse problem is derived [3]. We conclude by summarizing similarities and differences among inverse problems using classic and modifed transmission eigenvalues.

[1] Gintides D and Pallikarakis N, The inverse transmission eigenvalue problem for a discontinuous refractive index, Inverse Problems, 33, 2017.

[2] Gintides D, Pallikarakis N and Stratouras K, On the modified transmission eigenvalue problem with an artificial metamaterial background, Res. Math. Sci., 8, 2021, (special issue on transmission eigenvalues).

[3] Gintides D, Pallikarakis N and Stratouras K, Uniqueness of a spherically symmetric refractive index from modified transmission eigenvalues, Inverse Problems, 38, 2022.