标题： 一族各向异性的积分算子及其最大特征值的行为 显示英文标题

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

摘要： 我们研究了定义在$L^2(\mathbb R)$上的一族紧致积分算子$\mathbf K_\beta$，其核函数为 K_\beta (x, y) = \frac{1}{\pi}\frac{1}{1 + (x-y)^2 + \beta^2\Theta(x, y)}，依赖于参数$\beta >0$，其中$\Theta(x, y)$是一个次数为$\gamma\ge 1$的对称非负齐次函数。 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 $\mathbf A = |d/dx| + \Theta(x, x)/2$. 证明中起关键作用的事实是，$\mathbf K_\beta, \beta>0,$是正性改善的。 情况$\Theta(x, y) = (x^2 + y^2)^2$在文献中已被作为高温超导的简化模型进行过研究。 显示英文摘要

摘要： 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 $\Theta(x, y)$ 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 $\mathbf A = |d/dx| + \Theta(x, x)/2$. 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.