Universality in low-dimensional BCS theory

It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature Xi and the critical temperature T-c is (approximately) given by a universal constant, independent of the microscopic details of the fermionic interaction. This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density and high density. The goal of this short note is to extend the universal behavior to lower dimensions d = 1, 2 and give an exemplary proof in the weak coupling limit.