An anisotropic integral operator in high temperature superconductivity

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.