On new surface-localized transmission eigenmodes
Youjun Deng, Yan Li Jiang, Hongyu Liu, Kai Zhang · Inverse Problems and Imaging · 2021
Consider the transmission eigenvalue problem \begin{document}$ (\Delta+k^2\mathbf{n}^2) w = 0, \ \ (\Delta+k^2)v = 0\ \ \mbox{in}\ \ \Omega;\quad w = v, \ \ \partial_ u w = \partial_ u v\ \ \mbox{on} \ \partial\Omega. $\end{document} It is shown in [ 16 ] that there exists a sequence of eigenfunctions \begin{document}$ (w_m, v_m)_{m\in\mathbb{N}} $\end{document} associated with \begin{document}$ k_m\rightarrow \infty $\end{document} such that either \begin{document}$ \{w_m\}_{m\in\mathbb{N}} $\end{document} or \begin{document}$ \{v_m\}_{m\in\mathbb{N}} $\end{document} are surface-localized, depending on \begin{document}$ \mathbf{n}>1 $\end{document} or \begin{document}$ 01 $\end{document} or \begin{document}$ 0<\mathbf{n}<1 $\end{document} . Though our study is confined within the radial geometry, the construction is subtle and technical.