A block-Toeplitz approach to the spectrum of the Neumann-Poincaré operator on self-similar chains
Abstract
We characterise the spectrum of the Neumann-Poincaré operator acting on a mean-zero subspace of the Sobolev-Slobodeckij space Ws,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−1/20(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.
Related
- On the spectrum of the biharmonic Steklov operator with Neumann boundary conditions acting on forms of the unit Euclidean ball
- On the planar Neumann--Poincaré inverse spectral problem
- Hyponormal Toeplitz operators with operator valued normal symbols
- K-theoretic invariants of the Toeplitz operator system and its dual, and of all graph operator systems
- The Cesàro Operator, Rhaly operators, and the Toeplitz Algebra
BibTeX
@misc{ruiz2026blocktoeplitz,
title = {{A block-Toeplitz approach to the spectrum of the Neumann-Poincaré operator on self-similar chains}},
author = {Matias Ruiz},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30991},
eprint = {2608.30991},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30991},
}
RIS
TY - UNPB
TI - A block-Toeplitz approach to the spectrum of the Neumann-Poincaré operator on self-similar chains
AU - Matias Ruiz
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30991
UR - https://arxiv.org/abs/2608.30991
AB - 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.
ER -