Nonuniqueness of the inverse-scattering problem and the presence ofE=0bound states

Two different one-dimensional local potentials, $V_1$ and $V_2$, can have identical reflection coefficients $r_1\mathrm{}\left(k\right)=r_2\mathrm{}\left(k\right)\mathrm{}$ and identical transmission coefficients $t_1\mathrm{}\left(k\right)=t_2\mathrm{}\left(k\right)\mathrm{}$, as shown recently by Abraham, DeFacio, and Moses (ADM effect). We raise two questions concerning their example at $k=0\mathrm{}$. We demonstrate a third potential $V_3$ such that $r_3=r_2=r_1$ and $t_3=t_2=t_1$. We further demonstrate two other potentials, $V_4$ and $V_5$, which exhibit the ADM effect and for which the above two questions do not arise.