The paradoxical zero reflection at zero energy

TL;DR

This paper investigates the counterintuitive phenomenon where zero-energy particles can have zero reflection at certain potential wells, contrasting the usual total reflection, and explores the conditions and models where this occurs.

Contribution

It extends the concept of Half Bound State to 1D systems, showing zero reflection at zero energy occurs at critical potential parameters, supported by analytical and numerical models.

Findings

Zero reflection at zero energy occurs at critical potential parameters.

Analytical models demonstrate the phenomenon in square and exponential wells.

Numerical results support the critical behavior near specific parameters.

Abstract

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.