Eigenvalues of the Neumann-Poincare operator in dimension 3: Weyl's law and geometry

TL;DR

This paper studies the asymptotic distribution of eigenvalues of the Neumann-Poincare operator in three dimensions, revealing geometric influences and addressing the long-standing question of negative eigenvalues' finiteness.

Contribution

It provides detailed asymptotic formulas for positive and negative eigenvalues of the NP operator in 3D, advancing understanding of their geometric dependence and eigenvalue finiteness.

Findings

Eigenvalues follow Weyl's law with geometric coefficients.

Separate asymptotics for positive and negative eigenvalues are established.

Results contribute to the negative eigenvalues' finiteness problem.

Abstract

We consider the asymptotic properties of the eigenvalues of the Neumann-Poincare (NP) operator in three dimensions. The region $\Omega\subset \mathbb{R}^3$ is bounded by a compact surface $\Gamma=\partial \Omega$, with certain smoothness conditions imposed. The NP operator is defined by $ \mathcal{K}_{\Gamma}[\psi](\mathbf{x}):=\frac{1}{4\pi}\int_\Gamma\frac{\langle \mathbf{y}-\mathbf{x},\mathbf{n}(\mathbf{y})\rangle}{|\mathbf{x}-\mathbf{y}|^3}\psi(\mathbf{y})dS_{\mathbf{y}},$ where $dS_{\mathbf{y}}$ is the surface element and $\mathbf{n}(\mathbf{y})$ is the outer unit normal vector on $\Gamma$. The first-named author established earlier that the singular numbers $s_j(\mathcal{K}_{\Gamma})$ of $\mathcal{K}_{\Gamma}$ and the ordered moduli of its eigenvalues $\lambda_j(\mathcal{K}_{\Gamma})$ satisfy the Weyl law with coefficient expressed in geometric terms. Our main purpose here is to investigate the asymptotic behavior of positive and negative eigenvalues separately under the condition of infinite smoothness of the boundary $\Gamma$. These formulas are used, in particular, to obtain certain answers to the long-standing problem of the existence or finiteness of negative eigenvalues of $\mathcal{K}_{\Gamma}$.