The BCS functional of superconductivity and its mathematical properties

TL;DR

This paper reviews recent mathematical results on the BCS functional of superconductivity, including critical temperature analysis, external field effects, and derivation of the Ginzburg-Landau model from BCS theory.

Contribution

It provides a comprehensive overview of mathematical properties of the BCS functional, highlighting new insights into critical temperature and the connection to Ginzburg-Landau theory.

Findings

Analysis of critical temperature for various interaction potentials

Dependence of critical temperature on external fields

Derivation of Ginzburg-Landau model from BCS theory

Abstract

We review recent results concerning the mathematical properties of the Bardeen-Cooper-Schrieffer (BCS) functional of superconductivity, which were obtained in a series of papers partly in collaboration with R. Frank, E. Hamza, S. Naboko, and J.P. Solovej. Our discussion includes, in particular, an investigation of the critical temperature for a general class of interaction potentials, as well as a study of its dependence on external fields. We shall explain how the Ginzburg-Landau model can be derived from the BCS theory in a suitable parameter regime.