  We provide a new analytical and computational study of the transmission
eigenvalues with a conductive boundary condition. These eigenvalues are derived
from the scalar inverse scattering problem for an inhomogeneous material with a
conductive boundary condition. The goal is to study how these eigenvalues
depend on the material parameters in order to estimate the refractive index.
The analytical questions we study are: deriving Faber-Krahn type lower bounds,
the discreteness and limiting behavior of the transmission eigenvalues as the
conductivity tends to infinity for a sign changing contrast. We also provide a
numerical study of a new boundary integral equation for computing the
eigenvalues. Lastly, using the limiting behavior we will numerically estimate
the refractive index from the eigenvalues provided the conductivity is
sufficiently large but unknown.
