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

Abstract

We study the family of compact integral operators $\mathbf K_β$ 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 $β>0$, where $Θ(x, y)$ is a symmetric non-negative homogeneous function of degree $γ\ge 1$. The main result is the following asymptotic formula for the maximal eigenvalue $M_β$ of $\mathbf K_β$: M_β= 1 - λ_1 β^{\frac{2}{γ+1}} + o(β^{\frac{2}{γ+1}}), β\to 0, where $λ_1$ is the lowest eigenvalue of the operator $\mathbf A = |d/dx| + Θ(x, x)/2$. A central role in the proof is played by the fact that $\mathbf K_β, β>0,$ is positivity improving. The case $Θ(x, y) = (x^2 + y^2)^2$ has been studied earlier in the literature as a simplified model of high-temperature superconductivity.