Bose-Einstein condensates at equilibrium inside a pancake-shaped trap: dimensional cross-over in the scattering properties

Two-dimensionality of the scattering events in a Bose-Einstein condensate introduces a logarithmic dependence on density in the coupling constant entering a mean-field theory of the equilibrium density profile, which becomes dominant as the s-wave scattering length gets larger than the condensate thickness. We analyze quantitatively the role of the form of the coupling constant in determining the transverse profile of a condensate confined in a harmonic pancake-shaped trap at zero temperature. We trace the regions of experimentally accessible system parameters for which the cross-over between different dimensionality behaviors may become observable through in situ imaging of the condensed cloud with varying trap anisotropy and scattering length.