A decay estimate for the eigenvalues of the Neumann-Poincar\'{e} operator in two dimensions using the Grunsky coefficients

We investigate the decay property of the eigenvalues of the Neumann-Poincar'{e} operator in two dimensions. As is well-known, this operator admits only a sequence of eigenvalues that accumulates to zero as its spectrum for a bounded domain having $C^{1,\alpha }$ boundary with $\alpha\in (0,1)$\mathrm{.}Inthispaper,weshowthattheeigenvalue$\lambda_k$'s of the Neumann-Poincar'{e} operator ordered by size satisfy that $|\lambda_k| = O(k^{-p-\alpha+1/2})$foranarbitrarysimplyconnecteddomainhaving$C^{1+p,\alpha}$boundarywith$p\geq 0,~ \alpha\in(0,1)$and$p+\alpha>\frac{1}{2}$.