A review on the direct and inverse transmission eigenvalue problem for the spherically symmetric refractive index

The purpose of this review article is to explore the interior transmission eigenvalue problem on the unit ball of R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}<^>3$$\end{document}. Focusing on the direct and inverse transmission eigenvalue problems for the spherically symmetric refractive index, we present key findings without getting into many technical details. After the formulation of the eigenvalue problem, we outline theorems on existence and distribution of real and complex eigenvalues, asymptotic formulas, and uniqueness results for the inverse problem. Additionally, we provide a concise overview of the current bibliography on transmission eigenvalue problems, concluding with open questions in the spherically symmetric case.