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

2011
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Journal of Spectral Theory
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We study the family of compact integral operators K_β in L^2( R) with the kernel K_β(x, y) = 1/π1/1 + (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> 1. The main result is the following asymptotic formula for the maximal eigenvalue M_β of K_β: M_β = 1 - λ_1 β^2/γ+1 + o(β^2/γ+1), β→ 0, 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

doi:10.4171/jst/19
fatcat:3ol565wl7raghomsqaqbl25naq