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Extreme eigenvalues of an integral operator

We study the family of compact operators $B_α = V A_α V$, $α>0$ in $L^2(\mathbb R^d)$, $d\ge 1$, where $A_α$ is the pseudo-differential operator with symbol $a_α(\boldsymbolξ) = a(α\boldsymbolξ)$, and both functions $a$ and $V$ are real-valued and decay at infinity. We assume that $a$ and $V$ attain their maximal values $A_0>0$, $V_0>0$, only at $\boldsymbolξ= \mathbf 0$ and $\mathbf x = \mathbf 0$. We also assume that a(\boldsymbolξ) = &\ A_0 - Ψ_γ(\boldsymbolξ) + o(|\boldsymbolξ|^γ),\ |\boldsymbolξ|\to 0, V(\mathbf x) = &\ V_0 - Φ_β(\mathbf x) + o(|\mathbf x|^β),\ |\mathbf x|\to 0, with some functions $Ψ_γ(\boldsymbolξ)>0$, $\boldsymbolξ\not =\mathbf 0$ and $Φ_β(\mathbf x) >0$, $\mathbf x\not = \mathbf 0$ that are homogeneous of degree $γ>0$ and $β>0$ respectively. The main result is the following asymptotic formula for the eigenvalues $λ_α^{(n)}$ of the operator $B_α$ (arranged in descending order counting multiplicity) for fixed $n$ and $α\to 0$: λ_α^{(n)} = A_0V_0^2 - μ^{(n)} α^σ + o(α^σ), α\to 0, where $σ^{-1} = γ^{-1}+ β^{-1}$, and $μ^{(n)}$ are the eigenvalues (arranged in ascending order counting multiplicity) of the model operator $T$ with symbol $V_0^2Ψ_γ(\boldsymbolξ) + 2A_0 V_0 Φ_β(\mathbf x)$.