  The transmission eigenvalues corresponding to the half-line Schr\"odinger
equation with the general selfadjoint boundary condition is analyzed when the
potential is real valued, integrable, and compactly supported. It is shown that
a transmission eigenvalue corresponds to the energy at which the scattering
from the perturbed system agrees with the scattering from the unperturbed
system. A corresponding inverse problem for the recovery of the potential from
a set containing the boundary condition and the transmission eigenvalues is
analyzed, and a unique reconstruction of the potential is given provided one
additional constant is contained in the data set. The results are illustrated
with various explicit examples.
