发表机构
School of Computing and Mathematical Sciences, University of Leicester（莱斯特大学计算与数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对积聚于极限点的自相似链构成的非Lipschitz域，采用块托普利茨算子理论刻画Neumann-Poincaré算子的谱，推导了相关渐近谱分布与特征值计数公式，并通过实例验证结果。
AI 中文摘要
我们刻画了作用于Sobolev-Slobodeckij空间$W^{s,p}(\Gamma)$的均值为零子空间上的Neumann-Poincaré算子的谱，其中$\Gamma$是非Lipschitz域，由无限个不相交光滑域组成的自相似链构成，这些光滑域在一个极限点处积聚。我们利用该链的离散几何缩放特性，得到Neumann-Poincaré算子以及单层、双层算子的精确块托普利茨表示。对应的算子值符号完全依赖于单一缩放参数$\alpha=(d-1)/p-s$，且当$0<\alpha<d$时属于Wiener代数。这使我们能利用块托普利茨算子理论刻画该范围内的本质谱和Fredholm区域。在对应$\alpha = d/2$的能量空间$H^{-1/2}_0(\Gamma)$中，我们证明谱是实的，并通过算子值Wiener-Hopf分解刻画本质谱外的任何孤立特征值。此外，我们应用算子值Szegő极限定理推导大有限截断的渐近谱分布，并基于算子符号建立特征值计数公式。我们通过同心圆环链的显式解析计算和圆盘链的数值近似来说明这些结果。
英文摘要
We characterise the spectrum of the Neumann-Poincaré operator acting on a mean-zero subspace of the Sobolev-Slobodeckij space $W^{s,p}(Γ)$, where $Γ$ 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-Poincaré operator, as well as the single- and double-layer operators. The corresponding operator-valued symbol depends entirely on the single scaling parameter $α=(d-1)/p-s$ and belongs to the Wiener algebra whenever $0<α<d$. This allows us to exploit block-Toeplitz operator theory to characterise the essential spectrum and Fredholm regions within this regime. In the energy space $H^{-1/2}_0(Γ)$, corresponding to $α= d/2$, 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 Szegő 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.
