Inverse transmission eigenvalue problem for fixed angular momentum

Abstract

<p style='text-indent:20px;'>We consider the inverse problem of the interior transmission eigenvalue problem for spherically stratified media with fixed angular-momentum quantum number <inline-formula><tex-math id="M1">\begin{document}$ l $\end{document}</tex-math></inline-formula>. Under the case <inline-formula><tex-math id="M2">\begin{document}$ \int^1_0\sqrt{\rho(r)}{\rm d}r\leq1 $\end{document}</tex-math></inline-formula>, we show that a knowledge of all the transmission eigenvalues for <inline-formula><tex-math id="M3">\begin{document}$ l $\end{document}</tex-math></inline-formula> uniquely determine the refraction index <inline-formula><tex-math id="M4">\begin{document}$ \rho(r) $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M5">\begin{document}$ [0, 1] $\end{document}</tex-math></inline-formula>. And under the case <inline-formula><tex-math id="M6">\begin{document}$ \int^1_0\sqrt{\rho(r)}{\rm d}r>1 $\end{document}</tex-math></inline-formula>, we show that all the transmission eigenvalues for fixed <inline-formula><tex-math id="M7">\begin{document}$ l $\end{document}</tex-math></inline-formula> with a partial information of the refraction index <inline-formula><tex-math id="M8">\begin{document}$ \rho(r) $\end{document}</tex-math></inline-formula> imply the uniqueness of <inline-formula><tex-math id="M9">\begin{document}$ \rho(r) $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M10">\begin{document}$ [0, 1] $\end{document}</tex-math></inline-formula>.</p>