I propose a new family of nearly singular interior transmission eigenvalue problems for inverse scattering. For a known support of the inhomogeneity the eigenvalues allow to identify e.g. the piece-wise constant values of the refractive index. For an unknown support of the inhomogeneity the eigenvalues allow to construct an indicator function, I present an analysis for ideal settings and numerical examples for general cases.

The extension of sampling methods to the imaging of locally perturbed periodic layers [1] requires the analysis of interior transmission problems in unbounded waveguides. The resulting problem is no longer of Fredholm type and its study necessitates additional tools to those classically used for the case of bounded domains. We shall present a proof of well posdeness in the case of absorbing background using Floquet-Bloch transform and finite dimensional approximation with respect to the Floquet-Bloch variable. The analysis of absorption free problem and related transmission eigenvalues is an open problem that we shall also briefly discuss. We plan to additionally present the so-called differential sampling method where some specific single Floquet-Bloch variable transmission eigenvalue problems shows up. These problems had been adressed in the case of periodically distributed defects in [2].

[1] Y. Boukari, H. Haddar, N. Jenhani. Analysis of sampling methods for imaging a periodic layer and its defects, Inverse Problems, 2023.

[2] F. Cakoni, H. Haddar, T.-P. Nguyen. New interior transmission problem applied to a single Floquet-Bloch mode imaging of local perturbations in periodic media, Inverse Problems, 2018.

We give a short review of old and recent results on scatterers with transmission eigenvalues of infinite multiplicity, including transparent scatterers. Our examples include potentials from the Schwartz class and multipoint potentials of Bethe - Peierls type.

[1] P.G. Grinevich, R.G. Novikov. Russian Mathematical Surveys 77:1021-1028, 2022. https://doi.org/10.4213/rm10080e