Elastic scattering of a quantum matter-wave bright soliton on a barrier

We consider a one-dimensional matter-wave bright soliton, corresponding to the ground bound state of N bosonic particles of mass m having a binary attractive delta potential interaction on the open line. For a full N-body quantum treatment, we derive several results for the scattering of this quantum soliton on a short-range, bounded from below, external potential, restricting to the low energy, elastic regime where the centre-of-mass kinetic energy of the incoming soliton is lower than the internal energy gap of the soliton, that is the minimal energy required to extract particles from the soliton.