The scattering length at positive temperature

TL;DR

This paper introduces a positive temperature analogue of the scattering length, providing improved bounds applicable to infinite-range potentials, which are essential for understanding the critical temperature of dilute Bose gases.

Contribution

It extends and refines previous bounds on the scattering length at positive temperature, including infinite-range potentials, crucial for Bose gas critical temperature analysis.

Findings

Derived an improved upper bound on the positive temperature scattering length.

Extended the bound to include infinite-range potentials.

Enhanced understanding of the critical temperature in dilute Bose gases.

Abstract

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.