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

We consider the asymptotic properties of the eigenvalues of the Neumann–Poincaré ($\mathrm{NP}$) 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 $\mathrm{NP}$ operator $\mathcal{K}_{\Gamma}$, called often ‘the direct value of the double layer potential’, acting in $L^2(\Gamma)$, is defined by
\begin{equation*} \mathcal{K}_{\Gamma}[\psi](\mathbf{x}):=\frac{1}{4\pi}\int\limits_\Gamma\frac{\langle \mathbf{y}-\mathbf{x},\mathbf{n}(\mathbf{y})\rangle}{|\mathbf{x}-\mathbf{y}|^3}\psi(\mathbf{y})dS_{\mathbf{y}}, \end{equation*}
where $dS_{\mathbf{y}}$ is the surface element and $\mathbf{n}(\mathbf{y})$ is the outer unit normal on $\Gamma$. The first-named author proved in [27] 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
\begin{equation*} s_j(\mathcal{K}(\Gamma))\sim|\lambda_j(\mathcal{K}_{\Gamma})|\sim \{ \frac{3W(\Gamma)-2\pi\chi(\Gamma)}{128\pi}\}^{\frac12}j^{-\frac12}, \end{equation*}
under the condition that $\Gamma$ belongs to the class $C^{2, \alpha}$ with $\alpha>0$, where $W(\Gamma)$ and $\chi(\Gamma)$ denote, respectively, the Willmore energy and the Euler characteristic of the boundary surface $\Gamma$. Although the $\mathrm{NP}$ operator is not selfadjoint (and therefore no general relationships between eigenvalues and singular number exists), the ordered moduli of the eigenvalues of $\mathcal{K}_{\Gamma}$ satisfy the same asymptotic relation.
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 $\mathcal{K}_{\Gamma}$. A more sophisticated estimation allows us to give a natural extension of the Weyl law for the case of a smooth boundary.