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We consider a Bose gas with two-body Kac-like scaled interactions V $\gamma$(r) = $\gamma$ 3 v($\gamma$r) where v(x) is a given repulsive and integrablepotential, while $\gamma$ is a positive parameter which controls the range ofthe interactions and their amplitude at a distance r. Using the Hartree-Fockapproximation we find that, at finite non-zero temperatures, the Bose-Einsteincondensation is destroyed by the repulsive interactions when they aresufficiently long-range. More precisely, we show that for $\gamma$ sufficientlysmall but finite the off-diagonal part of the one-body density matrix alwaysvanishes at large distances. Our analysis sheds light on the coupling betweencritical correlations and long-range interactions, which might lead to thebreakdown of the off-diagonal long-range order even beyond the Hartree-Fockapproximation. Furthermore, our Hartree-Fock analysis shows the existence of athreshold value $\gamma$ 0 above which the Bose-Einstein condensation isrestored. Since $\gamma$ 0 is an unbounded increasing function of thetemperature this implies for a fixed $\gamma$, namely for a fixed scaledpotential, that a condensate cannot form above some critical temperaturewhatever the value of the density.

Hartree-Fock analysis of the effects of long-range interactions on the Bose-Einstein condensation