Energy dependent scattering and the Gross-Pitaevskii Equation in two   dimensional Bose-Einstein condensates
  We consider many-body effects on particle scattering in one, two and three dimensional Bose gases. We show that at zero temperature these effects can be modelled by the simpler two-body T-matrix evaluated off the energy shell. This is important in 1D and 2D because the two-body T-matrix vanishes at zero energy and so mean-field effects on particle energies must be taken into account to obtain a self-consistent treatment of low energy collisions. Using the off-shell two-body T-matrix we obtain the energy and density dependence of the effective interaction in 1D and 2D and the appropriate Gross-Pitaevskii equations for these dimensions. We present numerical solutions of the Gross-Pitaevskii equation for a 2D condensate of hard-sphere bosons in a trap. We find that the interaction strength is much greater in 2D than for a 3D gas with the same hard-sphere radius. The Thomas-Fermi regime is therefore approached at lower condensate populations and the energy required to create vortices is lowered compared to the 3D case. 
M.D. Lee, S.A. Morgan, M.J. Davis, and K. Burnett