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

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 $\mathrm{\Theta }(x,y)$ 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 $\mathbf A = |d/dx| + \Theta(x, x)/2$\mathrm{.}Acentralroleintheproofisplayedbythefactthat$\mathbf K_\beta, \beta>0,$ispositivityimproving\mathrm{.}Thecase$\Theta(x, y) = (x^2 + y^2)^2$ has been studied earlier in the literature as a simplified model of high-temperature superconductivity.