The paradoxical zero reflection at zero energy

Usually, the reflection probability $R(E)$ of a particle of zero energy incident on a potential which converges to zero asymptotically is found to be 1: $R(0)=1$. But earlier, a paradoxical phenomenon of zero reflection at zero energy ($R(0)=0$) has been revealed as a threshold anomaly. Extending the concept of Half Bound State (HBS) of 3D, here we show that in 1D when a symmetric (asymmetric) attractive potential well possesses a zero-energy HBS, $R(0)=0$ $(R(0)<<1)$. This can happen only at some critical values $q_c$ of an effective parameter $q$ of the potential well in the limit $E \rightarrow 0^+$. We demonstrate this critical phenomenon in two simple analytically solvable models which are square and exponential wells. However, in numerical calculations even for these two models $R(0)=0$ is observed only as extrapolation to zero energy from low energies, close to a precise critical value $q_c$. By numerical investigation of a variety of potential wells, we conclude that for a given potential well (symmetric or asymmetric), we can adjust the effective parameter $q$ to have a low reflection at a low energy.