JournalsjstVol. 1, No. 4pp. 443–460

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
A family of anisotropic integral operators and behavior of its maximal eigenvalue cover
Download PDF

Abstract

We study the family of compact integral operators Kβ{\bf K}_\beta in L2(R)L2(\mathbb R) with the kernel

Kβ(x,y)=1π11+(xy)2+β2Θ(x,y),K_\beta(x, y) = \frac{1}{\pi}\frac{1}{1 + (x-y)^2 + \beta^2\Theta(x, y)},

depending on the parameter β>0\beta >0, where Θ(x,y)\Theta(x, y) is a symmetric non-\hspace{0pt}negative homogeneous function of degree γ1\gamma\ge 1. The main result is the following asymptotic formula for the maximal eigenvalue MβM_\beta of Kβ\bf K_\beta:

Mβ=1λ1β2γ+1+o(β2γ+1),β0,M_\beta = 1 - \lambda_1 \beta^{\frac{2}{\gamma+1}} + o(\beta^{\frac{2}{\gamma+1}}),\quad \beta\to 0,

where λ1\lambda_1 is the lowest eigenvalue of the operator A=d/dx+Θ(x,x)/2\bf A = |d/dx| + \Theta(x, x)/2. A central role in the proof is played by the fact that Kβ,β>0,\bf K_\beta, \beta>0, is positivity improving. The case Θ(x,y)=(x2+y2)2\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