The scattering length at positive temperature

A positive temperature analogue of the scattering length of a potential $V$ can be defined via integrating the difference of the heat kernels of $-\Delta$ and $-\Delta + \frac 12 V$, with $\Delta$ the Laplacian. An upper bound on this quantity is a crucial input in the derivation of a bound on the critical temperature of a dilute Bose gas \cite{SU}. In \cite{SU} a bound was given in the case of finite range potentials and sufficiently low temperature. In this paper, we improve the bound and extend it to potentials of infinite range.