Levinson's theorem and reflectionless one-dimensional potentials

We studied two possible approaches to one-dimensional Levinson's theorem for Sch\"odinger equation. The first one, we restrict the 3-dimensional theorem. The other one, the theorem proposed by Dong, Ma and Klauss \cite{Dong}. We find failures in each approach for a one-dimensional reflectionless potential. In order to see this, we explicitly evaluate the phase shift using Schr\"odinger equation using solution procedure proposed by Jaffe \cite{Jaffe}.