Redundant poles of the $S$-matrix for the one dimensional Morse potential

We analyze the structure of the scattering matrix, $S(k)$, for the one dimensional Morse potential. We show that, in addition to a finite number of bound state poles and an infinite number of anti-bound poles, there exist an infinite number of redundant poles, on the positive imaginary axis, which do not correspond to either of the other types. This can be solved analytically and exactly. In addition, we obtain wave functions for all these poles and ladder operators connecting them. Wave functions for redundant state poles are connected via two different series. We also study some exceptional cases.