Bose condensates in a harmonic trap near the critical temperature
  The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local density approximation. Consistency of the theory for temperatures through the Bose condensation point requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length, increases the number of condensate atoms at each temperature, and raises the value of the transition temperature relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near the transition. Comparisons with the results of quantum Monte Carlo calculations and various local density approximations are presented, and experimental consequences are discussed. 
T. Bergeman, D. L. Feder, N. L. Balazs, and B. I. Schneider