We study the family of compact integral operators K_β in L^2( R) with the kernel K_β(x, y) = 1/π1/1 + (x-y)^2 + β^2Θ(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 M_β of K_β: M_β = 1 - λ_1 β^2/γ+1 + o(β^2/γ+1), β→ 0, where λ_1 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

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... _β, β>0, is positivity improving. The case Θ(x, y) = (x^2 + y^2)^2 has been studied earlier in the literature as a simplified model of high-temperature superconductivity.