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Interaction effects on atomic laboratory trapped Bose-Einstein condensates

We discuss the effect of inter-atoms interactions on the condensation temperature $T_c$ of an atomic laboratory trapped Bose-Einstein condensate. We show that, in the mean-field Hartree-Fock and semiclassical approximations, interactions produce a shift $ΔT_{c}/T_{c}^{0} \approx b_1 (a/λ_{T_c}) + b_2 (a/λ_{T_c})^2 + ψ[a/λ_{T_c}]$ with $a$ the s-wave scattering length, $λ_T$ the thermal wavelength and $ψ[a/λ_{T_c}]$ a non-analytic function such that $ψ[0] = ψ'[0] = ψ"[0] = 0$ and $|ψ"'[0]| = \infty$. Therefore, with no more assumptions than Hartree-Fock and semiclassical approximations, interaction effecs are perturbative to second order in $a/λ_{T_c}$ and the expected non-perturbativity of physical quantities at critical temperature appears only to third order. We compare this finding with different results by other authors, which are based on more than the Hartree-Fock and semiclassical approximations. Moreover, we obtain an analytical estimation for $b_2 \simeq 18.8$ which improves a previous numerical result. We also discuss how the discrepancy between $b_2$ and the empirical value of $b_2 = 46 \pm 5$ may be explained with no need to resort to beyond-mean field effects.