The critical temperature for the BCS equation at weak coupling

Abstract

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

The External Field Dependence of the BCS Critical Temperature

“…Thus, translation invariant BCS theory describes the relative coordinate of the Cooper pair wave function and GL theory the center of mass coordinate. The critical temperature T c in translation invariant BCS theory has been studied in detail in [5,12,14,15]. In this paper we investigate the critical temperature of the full BCS functional including (weak) external fields.…”

Weakly coupled bound states of Pauli operators

“…There is an enormous literature on weakly coupled eigenvalues and low energy behavior of Schrödinger operators from which we only mention [2,4,[6][7][8][9], the surveys [10,11] and the recent papers [12][13][14][15]. We emphasize that techniques from weakly coupled Schrödinger operators have also turned out to be useful in a non-linear context [16].…”

Microscopic Derivation of the Ginzburg-Landau Model

“…This linear criterion on the critical temperature was used to derive precise asymptotics of T c for weak coupling and/or low density, see [8]- [10].…”

“…The particle interaction V , on the other hand, varies on the microscopic scale. To take this into account, we replace the external fields A(x) and W (x) in the BCS functional (5) by (8) A…”