Analysis Seminar: "The essential spectrum of the Neumann-Poincare operator on a domain with corners"

Organizer/ Host: John McCarthy

Abstract: The study of the Neumann-Poincare (NP) operator (or the boundary double layer potential) of a domain dates back to Poincare and Carleman's doctoral dissertation, at the time serving as a prominent example in the abstract spectral theories proposed by Hilbert, Fredholm, and F. Riesz. Later, the NP operator was central in the development of the theory of singular integral operators. Very recently, the theory of new materials has revived the interest in the spectral properties of the NP operator, acting on the energy space of the domain. We use a classical similarity equivalence between the NP operator and the Ahlfors-Beurling transform of the domain to characterize the spectrum on a wedge in two variables. A localization argument combined with distortion estimates from conformal mapping leads to a complete description of the essential spectrum of the Neumann-Poincare operator on planar domains with corners.