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The transmission eigenvalues corresponding to the half-line Schr\"odingerequation with the general selfadjoint boundary condition is analyzed when thepotential is real valued, integrable, and compactly supported. It is shown thata transmission eigenvalue corresponds to the energy at which the scatteringfrom the perturbed system agrees with the scattering from the unperturbedsystem. A corresponding inverse problem for the recovery of the potential froma set containing the boundary condition and the transmission eigenvalues isanalyzed, and a unique reconstruction of the potential is given provided oneadditional constant is contained in the data set. The results are illustratedwith various explicit examples.

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