Characteristic discontinuity of the reflectivity curve in vicinity of total reflection in the case of an attractive slowly varying potential

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Abstract
The reflection of a plane wave on a potential which is zero at infinity may be calculated by a one dimensional Schrödinger equation (ħ 2/2m)ψ" + Eψ = Vψ, where E is the energy of the incidental wave and V the potential. When the energy is negative, the reflection is total (R(E) = 1) ; consequently we expect lim E →0+ R(E) to be equal to 1, the reflectivity curve being then continuous. But this is not always the case. When the potential is attractive and proportional to -1/(Z+Z0)α where 0 < α < 2, lim E→0+R (E ) does exist, but is strictly lower than 1. So we remark a discontinuity. The purpose of this study is to demonstrate the existence of this limit and to explicitly calculate the value