Scattering coefficients and gray-body factor for 1D BEC acoustic black holes: exact results

A complete set of exact analytic solutions to the mode equation is found in the region exterior to the acoustic horizon for a class of 1D Bose-Einstein condensate (BEC) acoustic black holes. From these, analytic expressions for the scattering coefficients and gray-body factor are obtained. The results are used to verify previous predictions regarding the behaviors of the scattering coefficients and gray-body factor in the low frequency limit.