Rigorous upper bound on the critical temperature of dilute Bose gases

We prove exponential decay of the off-diagonal correlation function in the two-dimensional homogeneous Bose gas when a(2)rho is small and the temperature T satisfies T4 pi rho ln vertical bar ln(a(2)rho)vertical bar. Here, a is the scattering length of the repulsive interaction potential and rho is the density. To the leading order in a(2)rho, this bound agrees with the expected critical temperature for superfluidity. In the three-dimensional Bose gas, exponential decay is proved when T-(TcTc(0))-T-(0)>5a rho(1/3), where T-c((0)) is the critical temperature of the ideal gas. While this condition is not expected to be sharp, it gives a rigorous upper bound on the critical temperature for Bose-Einstein condensation.