# 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 $K_{β}$ in $L2(R)$ with the kernel

depending on the parameter $β>0$, where $Θ(x,y)$ is a symmetric non-\hspace{0pt}negative homogeneous function of degree $γ≥1$. The main result is the following asymptotic formula for the maximal eigenvalue $M_{β}$ of $K_{β}$:

where $λ_{1}$ is the lowest eigenvalue of the operator $A=∣d/dx∣+Θ(x,x)/2$. A central role in the proof is played by the fact that $K_{β},β>0,$ 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.

## 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