The critical temperature for the BCS equation at weak coupling

TL;DR

This paper rigorously analyzes how the critical temperature in the BCS equation behaves as the interaction strength diminishes, providing conditions for the existence of solutions at all coupling levels.

Contribution

It offers a rigorous asymptotic analysis of the BCS critical temperature at weak coupling and establishes conditions on the potential for solution existence.

Findings

Derived asymptotic behavior of critical temperature as coupling approaches zero

Established necessary and sufficient conditions on the potential for solutions

Provided a rigorous mathematical framework for weak coupling analysis

Abstract

For the BCS equation with local two-body interaction $\lambda V(x)$, we give a rigorous analysis of the asymptotic behavior of the critical temperature as $\lambda \to 0$. We derive necessary and sufficient conditions on $V(x)$ for the existence of a non-trivial solution for all values of $\lambda>0$.