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

### Boris S. Mityagin

Ohio State University, Columbus, USA### Alexander V. Sobolev

University College London, UK

## Abstract

We study the family of compact integral operators ${\bf K}_\beta$ 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 $\bf K_\beta$:

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.

## Cite this article

Boris S. Mityagin, Alexander V. Sobolev, A family of anisotropic integral operators and behavior of its maximal eigenvalue. J. Spectr. Theory 1 (2011), no. 4, pp. 443–460

DOI 10.4171/JST/19