Bose–Einstein Condensation in a Harmonic Trap: Effect of Interactions on Tc

For a harmonically trapped dilute Bose gas with uniformly repulsive interactions which is assumed to satisfy a certain condition on the extensivity of fluctuations, I find on upper bound on the condensate fraction f. If BEC is defined by the condition that f>const.N−α, α<1/2, I argue that in the limit N→∞, Vo→0, NVo→const. where Vo is the space integral of the potential, the interactions cannot increase the critical temperature over that of the noninteracting gas.