  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 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.
