  In this paper, we present a Spectral-Galerkin Method to approximate the
zero-index transmission eigenvalues with a conductive boundary condition. This
is a new eigenvalue problem derived from the scalar inverse scattering problem
for an isotropic media with a conductive boundary condition. In our analysis,
we will consider the equivalent fourth-order eigenvalue problem where we
establish the convergence when the approximation space is the span of finitely
many Dirichlet eigenfunctions for the Laplacian. We establish the convergence
rate of the spectral approximation by appealing to Weyl's law. Numerical
examples for computing the eigenvalues and eigenfunctions for the unit disk and
unit square are presented. Lastly, we provide a method for estimating the
refractive index assuming the conductivity parameter is either sufficiently
large or small but otherwise unknown.
