Quantum reflection and Liouville transformations from wells to walls
  Liouville transformations map in a rigorous manner one Schr\"odinger equation into another, with a changed scattering potential. They are used here to transform quantum reflection of an atom on an attractive well into reflection of the atom on a repulsive wall. While the scattering properties are preserved, the corresponding semiclassical descriptions are completely different. A quantitative evaluation of quantum reflection probabilities is deduced from this method. 
Gabriel Dufour, Romain Gu\'erout, Astrid Lambrecht, Serge Reynaud
quant-ph