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

Abstract

We study the family of compact integral operators ${\bf K}\_\beta$ in $L2(\mathbb R)$ 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-\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 $\bf K\_\beta$: $$ M\_\beta = 1 - \lambda\_1 \beta^{\frac{2}{\gamma+1}} + o(\beta^{\frac{2}{\gamma+1}}),\quad \beta\to 0, $$ where $\lambda\_1$ is the lowest eigenvalue of the operator $\bf A = |d/dx| + \Theta(x, x)/2$. A central role in the proof is played by the fact that $\bf K\_\beta, \beta>0,$ 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.