The paradoxical zero reflection at zero energy

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 → 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. * Electronic address: 1:zahmed@barc.gov.in, 2:

Low reflection at zero or low-energies in the well-barrier scattering potentials

“…We find that, if q is an integer then ψ n=q (x) becomes the critical solitary HBS ψ * (x) of the Scarf II potential (5). Here y(x) = sinh x and P α,β n (z) are Jacobi polynomials.…”

“…A. Double Dirac delta well barrier potential DDDP [1][2][3][4][5] is depicted in Fig. 1(a) by solid line and it is written as…”

“…This observation is also intuitive, for a zero-energy particle the tunnel effect is negligible such that the transmission probability is close to zero. A paradoxical phenomenon that R(0) = 0 or R(0) < 1 has been revealed and proved as a threshold anomaly [1][2][3][4][5] for a potential which is at the threshold of binding a state at E = 0. This paradoxical result may be understood in terms of wave packet scattering from an attractive potential.…”

“…Recently, the paradoxical reflection has been shown to exist in the most simple square and exponential wells [5] which are symmetric. Low reflection at low energies has also been discussed.…”

“…In a crucial comment to Ref. [5], van Dijk and Nogami [6] emphasized the occurrence of R(0) < 1 in non-symmetric DDDP model. Interestingly, we found this comment [6] useful to visualize asymmetric DDDP (λ λ < 0 in [6]) as a well barrier potential.…”