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Enhanced Superconductivity at a Corner for the Linear BCS Equation

Barbara Roos, Robert Seiringer

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.

Enhanced Superconductivity at a Corner for the Linear BCS Equation

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 . Furthermore, we prove that the relative difference of the critical temperatures vanishes in the weak coupling limit.
Paper Structure (27 sections, 21 theorems, 203 equations, 1 figure)

This paper contains 27 sections, 21 theorems, 203 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof strategy for Theorem \ref{['thm1']}
  3. Basic properties of $H_T^{\Omega_1}$ and $H_T^{\Omega_2}$
  4. Proof of Lemma \ref{['t2geqt1']}
  5. Proof of Lemma \ref{['lea:etaex']}
  6. Proof of Lemma \ref{['lea:H1ev']}
  7. Regularity and asymptotic behavior of the half-space ground state
  8. Proof of Lemma \ref{['etato0']}
  9. Proof of Lemma \ref{['lea:phi_infty']}
  10. Proof of Lemma \ref{['lea:Vphi_reg.1']}
  11. Proof of Lemma \ref{['lea:cont_bd']}
  12. Proof of Lemma \ref{['lea:eps_to_0']}
  13. Proof of \ref{['l0_pf']}:
  14. Proof of \ref{['l1_pf']}:
  15. Proof of \ref{['l2_pf']}:
  16. ...and 12 more sections

Key Result

Lemma 1.1

Let $\lambda,T>0$ and $V\in L^{t}({\mathbb{R}}^2)$ for some $t>1$. Then $\inf \sigma(H_T^{\Omega_2})\leq \inf \sigma(H_T^{\Omega_1})$.

Figures (1)

  • Figure 1: Sketch of the (anti)symmetric extension of a function $\psi$ defined on the upper right quadrant in the $(r_1,z_1)$-coordinates. The extension is defined by mirroring along the $x_1$ and $y_1$-axes and multiplying by $- 1$ for Dirichlet boundary conditions.

Theorems & Definitions (41)

  • Lemma 1.1
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Lemma 1.7
  • Lemma 1.8
  • Lemma 1.9
  • Lemma 1.10
  • Remark 1.11
  • ...and 31 more