A family of anisotropic integral operators and behaviour of its maximal eigenvalue

الأصل · EN

We study the family of compact integral operators Kβ in L²(R) with the kernel Kβ(x, y) = 1π1/1 + (x-y)² + β²Θ(x, y), depending on the parameter β>0, where Θ(x, y) is a symmetric non-negative homogeneous function of degree γ≥ 1. The main result is the following asymptotic formula for the maximal eigenvalue Mβ of Kβ: Mβ= 1 - λ₁ β²/γ⁺¹ + o(β²/γ⁺¹), β→ 0, where λ₁ is the lowest eigenvalue of the operator A = |d/dx| + Θ(x, x)/2. A central role in the proof is played by the fact that Kβ, β>0, is positivity improving. The case Θ(x, y) = (x² + y²)² has been studied earlier in the literature as a simplified model of high-temperature superconductivity.