Estimates of complex eigenvalues and an inverse spectral problem for the transmission eigenvalue problem

• Main Authors: Xiao-Chuan Xu, Chuan-Fu Yang, Sergey Buterin, Vjacheslav Yurko
• Format: Article
• Language: English
• Published: University of Szeged 2019-06-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: transmission eigenvalue problem; scattering theory; complex eigenvalue; inverse spectral problem
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=7380

This work deals with the interior transmission eigenvalue problem: $y'' + {k^2}\eta \left( r \right)y = 0$ with boundary conditions ${y\left( 0 \right) = 0 = y'\left( 1 \right)\frac{{\sin k}}{k} - y\left( 1 \right)\cos k},$ where the function $\eta(r)$ is positive. We obtain the asymptotic distribution of non-real transmission eigenvalues under the suitable assumption on the square of the index of refraction $\eta(r)$. Moreover, we provide a uniqueness theorem for the case $\int_0^1\sqrt{\eta(r)}dr>1$, by using all transmission eigenvalues (including their multiplicities) along with a partial information of $\eta(r)$ on the subinterval. The relationship between the proportion of the needed transmission eigenvalues and the length of the subinterval on the given $\eta(r)$ is also obtained.