Enhanced Superconductivity at a Corner for the Linear BCS Equation

TL;DR

This paper investigates how geometric confinement in two-dimensional superconductors affects the critical temperature, showing that corners enhance superconductivity at weak coupling, with differences diminishing as coupling weakens.

Contribution

It provides a rigorous comparison of critical temperatures in different geometries for the linear BCS equation, highlighting the impact of corners on superconductivity.

Findings

Critical temperature is higher in a quadrant than in a half-space.

Critical temperature in a half-space exceeds that in the full plane.

The relative difference in critical temperatures vanishes at weak coupling.

Abstract

We consider the critical temperature for superconductivity, defined via the linear BCS equation. We prove that at weak coupling the critical temperature for a sample confined to a quadrant in two dimensions is strictly larger than the one for a half-space, which in turn is strictly larger than the one for $\mathbb{R}^2$. Furthermore, we prove that the relative difference of the critical temperatures vanishes in the weak coupling limit.