On new surface-localized transmission eigenmodes

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_\nu w = \partial_\nu 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 $\end{document} or \begin{document}$ 0<\mathbf{n}<1 $\end{document}. In this paper, we discover a new type of surface-localized transmission eigenmodes by constructing a sequence of transmission eigenfunctions \begin{document}$ (w_m, v_m)_{m\in\mathbb{N}} $\end{document} associated with \begin{document}$ k_m\rightarrow \infty $\end{document} such that both \begin{document}$ \{w_m\}_{m\in\mathbb{N}} $\end{document} and \begin{document}$ \{v_m\}_{m\in\mathbb{N}} $\end{document} are surface-localized, no matter \begin{document}$ \mathbf{n}>1 $\end{document} or \begin{document}$ 0<\mathbf{n}<1 $\end{document}. Though our study is confined within the radial geometry, the construction is subtle and technical.