A block-Toeplitz approach to the spectrum of the Neumann-Poincar\'e operator on self-similar chains
We characterise the spectrum of the Neumann-Poincar\'e operator acting on a mean-zero subspace of the Sobolev-Slobodeckij space $W^{s,p}(\Gamma)$, where $\Gamma$ is a non-Lipschitz domain, consisting of an infinite self-similar chain of disjoint smooth domains accumulating at a limit point. We exploit the discrete geometric scaling of the chain to obtain an exact block-Toeplitz representation of the Neumann-Poincar\'e operator, as well as the single- and double-layer operators. The corresponding operator-valued symbol depends entirely on the single scaling parameter $\alpha=(d-1)/p-s$ and belongs to the Wiener algebra whenever $0<\alpha<d$. This allows us to exploit block-Toeplitz operator theory to characterise the essential spectrum and Fredholm regions within this regime. In the energy space $H^{-1/2}_0(\Gamma)$, corresponding to $\alpha = d/2$, we prove the spectrum is real and characterise any isolated eigenvalues outside the essential spectrum through an operator-valued Wiener-Hopf factorisation. Furthermore, we apply an operator-valued Szeg\H{o} limit theorem to derive the asymptotic spectral distribution for large finite truncations and establish an eigenvalue counting formula based on the operator symbol. We illustrate these results through explicit analytical computations for a chain of concentric annuli and numerical approximations for a chain of disks.