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

Abstract

We study the family of compact integral operators ${\mathbf{K}}_{\beta }$ in $L^2(\mathbb{R})$ with the kernel K_β(x, y) = \frac{1}π\frac{1}{1 + (x-y)^2 + β^2Θ(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_β= 1 - λ_1 β^{\frac{2}{γ+1}} + o(β^{\frac{2}{γ+1}}), β\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.

16 pages