Important Issues on Spectral Properties of a Transmission Eigenvalue Problem

Nowadays, inverse scattering is an important field of interest for many mathematicians who deal with partial differential equations theory, and the research in inverse scattering is in continuous progress. There are many problems related to scattering by an inhomogeneous media. Here, we study the transmission eigenvalue problem corresponding to a new scattering problem, where boundary conditions differ from any other interior problem studied previously. more specifically, instead of prescribing the difference Cauchy data on the boundary which is the classical form of the problem, we consider the case when the difference of the trace of the fields is proportional to the normal derivative of the field. Typical concerns related to TEP (transmission eigenvalue problem) are Fredholm property and solvability, the discreteness of the transmission eigenvalues, and their existence. In this article, we provide answers for all these concerns in a given interior transmission problem for an inhomogeneous media. We use the variational method and a very important theorem on the existence of transmission eigenvalues to arrive at the conclusion of the existence of the transmission eigenvalues.

On new surface-localized transmission eigenmodes

<p style='text-indent:20px;'>Consider the transmission eigenvalue problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>It is shown in [<xref ref-type="bibr" rid="b16">16</xref>] that there exists a sequence of eigenfunctions <inline-formula><tex-math id="M1">\begin{document}$ (w_m, v_m)_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> associated with <inline-formula><tex-math id="M2">\begin{document}$ k_m\rightarrow \infty $\end{document}</tex-math></inline-formula> such that either <inline-formula><tex-math id="M3">\begin{document}$ \{w_m\}_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M4">\begin{document}$ \{v_m\}_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> are surface-localized, depending on <inline-formula><tex-math id="M5">\begin{document}$ \mathbf{n}&gt;1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M6">\begin{document}$ 0&lt;\mathbf{n}&lt;1 $\end{document}</tex-math></inline-formula>. In this paper, we discover a new type of surface-localized transmission eigenmodes by constructing a sequence of transmission eigenfunctions <inline-formula><tex-math id="M7">\begin{document}$ (w_m, v_m)_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> associated with <inline-formula><tex-math id="M8">\begin{document}$ k_m\rightarrow \infty $\end{document}</tex-math></inline-formula> such that both <inline-formula><tex-math id="M9">\begin{document}$ \{w_m\}_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \{v_m\}_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> are surface-localized, no matter <inline-formula><tex-math id="M11">\begin{document}$ \mathbf{n}&gt;1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M12">\begin{document}$ 0&lt;\mathbf{n}&lt;1 $\end{document}</tex-math></inline-formula>. Though our study is confined within the radial geometry, the construction is subtle and technical.</p>

A new approach to the inverse discrete transmission eigenvalue problem

<p style='text-indent:20px;'>A discrete analog is considered for the inverse transmission eigenvalue problem, having applications in acoustics. We provide a well-posed inverse problem statement, develop a constructive procedure for solving this problem, prove uniqueness of solution, global solvability, local solvability, and stability. Our approach is based on the reduction of the discrete transmission eigenvalue problem to a linear system with polynomials of the spectral parameter in the boundary condition.</p>