  The paper concerns the isotropic interior transmission eigenvalue (ITE)
problem. This problem is not elliptic, but we show that, using the
Dirichlet-to-Neumann map, it can be reduced to an elliptic one. This leads to
the discreteness of the spectrum as well as to certain results on possible
location of the transmission eigenvalues. If the index of refraction
$\sqrt{n(x)}$ is real, we get a result on the existence of infinitely many
positive ITEs and the Weyl type lower bound on its counting function. All the
results are obtained under the assumption that $n(x)-1$ does not vanish at the
boundary of the obstacle or it vanishes identically, but its normal derivative
does not vanish at the boundary. We consider the classical transmission problem
as well as the case when the inhomogeneous medium contains an obstacle. Some
results on the discreteness and localization of the spectrum are obtained for
complex valued $n(x)$.
