Extreme eigenvalues of an integral operator

Abstract

We study the family of compact operators $B_{\alpha }=V{A}_{\alpha }V$, $\alpha >0$ in $L^2({\mathbb{R}}^d)$, $d\ge 1$, where $A_{\alpha }$ is the pseudo-differential operator with symbol $a_{\alpha }(\mathbf{\xi })=a(\alpha \mathbf{\xi })$, 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 $\mathbf{\xi }=0$ and $\mathbf{x}=0$. 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 ${\Psi }_{\gamma }(\mathbf{\xi })>0$, $\mathbf{\xi }\ne 0$ and ${\Phi }_{\beta }(\mathbf{x})>0$, $\mathbf{x}\ne 0$ that are homogeneous of degree $\gamma >0$ and $\beta >0$ respectively. The main result is the following asymptotic formula for the eigenvalues ${\lambda }_{\alpha }^{(n)}$ of the operator $B_{\alpha }$ (arranged in descending order counting multiplicity) for fixed $n$ and $\alpha \to 0$: λ_α^{(n)} = A_0V_0^2 - μ^{(n)} α^σ + o(α^σ), α\to 0, where ${\sigma }^{-1}={\gamma }^{-1}+{\beta }^{-1}$, and ${\mu }^{(n)}$ are the eigenvalues (arranged in ascending order counting multiplicity) of the model operator $T$ with symbol $V_0^2{\Psi }_{\gamma }(\mathbf{\xi })+2A_0{V}_0{\Phi }_{\beta }(\mathbf{x})$.

11 pages