  We study the family of compact integral operators $\mathbf K_\beta$ in
$L^2(\mathbb R)$ with the kernel K_\beta(x, y) = \frac{1}{\pi}\frac{1}{1 +
(x-y)^2 + \beta^2\Theta(x, y)}, depending on the parameter $\beta >0$, where
$\Theta(x, y)$ is a symmetric non-negative homogeneous function of degree
$\gamma\ge 1$. The main result is the following asymptotic formula for the
maximal eigenvalue $M_\beta$ of $\mathbf K_\beta$: M_\beta = 1 - \lambda_1
\beta^{\frac{2}{\gamma+1}} + o(\beta^{\frac{2}{\gamma+1}}), \beta\to 0, where
$\lambda_1$ is the lowest eigenvalue of the operator $\mathbf A = |d/dx| +
\Theta(x, x)/2$. A central role in the proof is played by the fact that
$\mathbf K_\beta, \beta>0,$ is positivity improving. The case $\Theta(x, y) =
(x^2 + y^2)^2$ has been studied earlier in the literature as a simplified model
of high-temperature superconductivity.
