Liouville transformations and quantum reflection

Liouville transformations of Schr"odinger equations preserve the scattering amplitudes while changing the effective potential. We discuss the properties of these gauge transformations and introduce a special Liouville gauge which allows one to map the problem of quantum reflection of an atom on an attractive Casimir-Polder well into that of reflection on a repulsive wall. We deduce a quantitative evaluation of quantum reflection probabilities in terms of the universal probability which corresponds to the solution of the $V_4=-C_4\mathrm{/}{z}^4$ far-end Casimir-Polder potential.