Quantum reflection: The invisible quantum barrier
  We construct the invisible quantum barrier which represents the phenomenon of quantum reflection using the available data. We use the Abel equation to invert the data. The resulting invisible quantum barrier is double-valued in both axes. We study this invisible barrier in the case of atom and Bose-Einstein Condensate reflection from a solid silicon surface. A time-dependent, one-spatial dimension Gross-Pitaevskii equation is solved for the BEC case. We found that the BEC behaves very similarly to the single atom except for size effects, which manifest themselves in a maximum in the reflectivity at small distances from the wall. The effect of the atom-atom interaction on the BEC reflection and correspondingly on the invisible barrier is found to be appreciable at low velocities and comparable to the finite size effect. The trapping of ultracold atom or BEC between two walls is discussed. 
J. X. de Carvalho, M. S. Hussein, and Weibin Li