Abstract:We study the relation between the Bogoliubov-de Gennes equation and the Ginzburg-Landau equation for a BCS model without external fields. While previous rigorous derivations of Ginzburg-Landau theory from BCS theory have focused on energies and minimizers, here we consider arbitrary critical points in the relevant energy regime. For temperatures close to the critical temperature, we prove that every sufficiently small solution of the BdG equation admits an asymptotic factorization into a microscopic Cooper-pair profile and a macroscopic order parameter. The latter satisfies the Ginzburg-Landau equation up to an error that vanishes in the scaling limit. Our analysis relies on a Birman-Schwinger reformulation of the BdG equation, a Lyapunov-Schmidt type reduction, and semiclassical estimates at low regularity.
| Comments: | 41 pages |
| Subjects: | Analysis of PDEs (math.AP); Mathematical Physics (math-ph) |
| Cite as: | arXiv:2605.26008 [math.AP] |
| (or arXiv:2605.26008v1 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.26008 arXiv-issued DOI via DataCite (pending registration) |
From: Rupert Frank [view email]
[v1]
Mon, 25 May 2026 16:27:25 UTC (43 KB)
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