The Dilute Bose-Einstein Condensate with Large Scattering Length

We study a dilute Bose gas of atoms whose scattering length a is large compared to the range of their interaction. We calculate the energy density of the homogeneous Bose-Einstein condensate to second order in the low-density expansion, expressing it in terms of a and a second parameter Lambda_* that determines the low-energy observables in the 3-body sector. The second-order correction to the energy density has a small imaginary part that reflects the instability due to 3-body recombination. In the case of a trapped Bose-Einstein condensate with large negative scattering length, we calculate the coefficient of the 3-body mean-field term in the energy density in terms of a and Lambda_*. It can be very large if there is an Efimov state near threshold.