An anisotropic integral operator in high temperature superconductivity

TL;DR

This paper rigorously analyzes an anisotropic integral operator in high temperature superconductivity, confirming the asymptotic behavior of the critical temperature shift conjectured by previous models.

Contribution

It provides a rigorous mathematical proof of the asymptotic behavior of the eigenvalue problem related to the superconductivity model, using the Additive Uncertainty Principle.

Findings

Proves the asymptotic behavior of the eigenvalue shift as a approaches zero.

Validates the conjectured relation between eigenvalues and critical temperature.

Employs the Additive Uncertainty Principle in the analysis.

Abstract

A simplified model in superconductivity theory studied by P. Krotkov and A. Chubukov \cite{KC1,KC2} led to an integral operator $K$ -- see (1), (2). They guessed that the equation $E_0(a,T)=1$ where $E_0$ is the largest eigenvalue of the operator $K$ has a solution $T(a)=1-\tau(a)$ with $\tau (a) \sim a^{2/5}$ when $a$ goes to 0. $\tau(a)$ imitates the shift of critical (instability) temperature. We give a rigorous analysis of an anisotropic integral operator $K$ and prove the asymptotic ($*$) -- see Theorem 8 and Proposition 10. Additive Uncertainty Principle (of Landau-Pollack-Slepian [SP], \cite{LP1,LP2}) plays important role in this analysis.