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

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:=\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 $\mathcal{K}{\Gamma}$.