Extreme eigenvalues of an integral operator

TL;DR

This paper analyzes the asymptotic behavior of the extreme eigenvalues of a family of integral operators with specific decay properties, revealing their relation to a model operator as a parameter approaches zero.

Contribution

It provides a novel asymptotic formula for eigenvalues of a class of integral operators with decaying symbols and potentials, connecting them to a model operator's spectrum.

Findings

Eigenvalues follow a specific asymptotic expansion as the parameter approaches zero.

The leading term of the eigenvalues is determined by the maximal values of the functions involved.

The eigenvalues are related to the spectrum of a model operator with a combined symbol.

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 $a_{\alpha}(\boldsymbol\xi) = a(\alpha\boldsymbol\xi)$, 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 $\boldsymbol\xi = \mathbf 0$ and $\mathbf x = \mathbf 0$. We also assume that a(\boldsymbol\xi) = &\ A_0 - \Psi_{\gamma}(\boldsymbol\xi) + o(|\boldsymbol\xi|^{\gamma}),\ |\boldsymbol\xi|\to 0, V(\mathbf x) = &\ V_0 - \Phi_{\beta}(\mathbf x) + o(|\mathbf x|^{\beta}),\ |\mathbf x|\to 0, with some functions $\Psi_{\gamma}(\boldsymbol\xi)>0$, $\boldsymbol\xi\not =\mathbf 0$ and $\Phi_{\beta}(\mathbf x) >0$, $\mathbf x\not = \mathbf 0$ 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 $\alpha\to 0$: \lambda_{\alpha}^{(n)} = A_0V_0^2 - \mu^{(n)} \alpha^{\sigma} + o(\alpha^{\sigma}), \alpha\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 $V_0^2\Psi_{\gamma}(\boldsymbol\xi) + 2A_0 V_0 \Phi_{\beta}(\mathbf x)$.