Title: Spectral Analysis of the Neumann–Poincaré Operator in Thin Planar Domains
Abstract: This talk examines the spectral characteristics of the Neumann–Poincaré (NP) operator in doubly connected, thin planar domains, focusing on theoretical aspects of spectral structure in this unique geometry. By constructing conformal mappings and employing Faber polynomials and Grunsky coefficients, we capture the essential spectral behavior of the NP operator. This analysis includes semi-infinite matrix representations and eigenvalue estimation techniques, particularly leveraging the Gershgorin circle theorem to characterize the spectrum’s limits. As the domain becomes infinitesimally thin, we show that the NP operator’s spectrum converges to the interval [−1/2,1/2], which reveals important insights into the resonance phenomena in thin structures. This work is a collaboration with Mikyoung Lim (KAIST) and Stephen P. Shipman (LSU).