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

Abstract

This work deals with the interior transmission eigenvalue problem: $y^{\prime \prime }+k^2\eta \left(r\right)y=0$ with boundary conditions $y\left(0\right)=0=y^{\prime }\left(1\right)\frac{\sin k}{k}-y\left(1\right)\cos k,$ where the function $\eta (r)$ is positive. We obtain the asymptotic distribution of non-real transmission eigenvalues under the suitable assumption for 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.

Cited by in corpus (2)

• On a regularization approach to the inverse transmission eigenvalue problem
• On the direct and inverse transmission eigenvalue problems for the Schrodinger operator on the half line