Transmission eigenvalues for the selfadjoint Schr\"odinger operator on the half line

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.