I was in my undergraduate QM lecture and we just finished with the Dirac barrier. My question is as follows: We know that the Dirac well’s complete set of solutions requires one bound state and an infinite set of scattering states (plane waves). The solutions to the Dirac barrier are a only a set of scattering states (plane waves again). How can both sets of solutions be complete? I asked this question in lecture and was told that no one-to-one correspondence exists between the two sets of plane waves, so the comparison is improper, but I’ve been unable to find anything else on the subject.