An Anisotropic Integral Operator in High Temperature Superconductivity

A simplified model in superconductivity theory studied by P. Krotkov and A. Chubukov [KC1, KC2] led to an integral operator K — see (1), (2). They guessed that the equation E0(a, T ) = 1 where E0 is the largest eigenvalue of the operator K has a solution (∗) T (a) = 1− τ(a) with τ(a) ∼ a when a goes to 0. τ(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], [LP1, LP2]) plays important role in this analysis. 0. Many models of high temperature superconductivity [LV] lead to the family of integral operators with anisotropic kernels. Structure and spectral analysis of these operators could be difficult and quite interesting because standard analytical methods (perturbation theory, Fourier transform, etc.) do not necessarily help us. P. Krotkov and A. Chubukov [KC1, KC2] [see [KC2], section B.2, (46)–(60)] simplified one of local Eliashberg gap equations by dropping the Matsubara frequency summation and came to the operator in L(R) (1) K ≡ KaT : f(x) → ∫ ∞