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

Electronic Journal of Qualitative Theory of Differential Equations

Chuan-Fu Yang¹  Vjacheslav Yurko²  Sergey Buterin²  Xiao-Chuan Xu³ 

[1] Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, Jiangsu, P.R. China;Department of Mathematics, Saratov State University, Saratov, Russia;School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, P.R. China,;

关键词: transmission eigenvalue problem;    scattering theory;    complex eigenvalue;    inverse spectral problem;   

DOI  :  10.14232/ejqtde.2019.1.38

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown