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

Abstract

We study the family of compact integral operators ${\mathbf{K}}_{\beta }$ in $L2(\mathbb{R})$ with the kernel

depending on the parameter $\beta >0$, where $\Theta (x,y)$ is a symmetric non-\hspace{0pt}negative homogeneous function of degree $\gamma \ge 1$. The main result is the following asymptotic formula for the maximal eigenvalue $M_{\beta }$ of ${\mathbf{K}}_{\beta }$:

where ${\lambda }_1$ is the lowest eigenvalue of the operator $\mathbf{A}=|d/dx|+\mathbf{\Theta }(\mathbf{x},\mathbf{x})/2$. 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.