Karl-Mikael Perfekt (U. of Reading)
The Neumann-Poincare (NP) operator (or the double layer potential) has classically been used as a tool to solve the Dirichlet and Neumann problems of a domain. However, it also serves as a prominent example in non-self adjoint spectral theory, due to its unexpected behaviour for domains with singularities. Recently, questions from materials science have revived interest in the spectral properties of the NP operator on domains with corners, edges, and conical points. This surge in attention is owed to the connection with resonances of transmission/scattering problems used to model surface plasmons in nanoparticles. I aim to give an overview of recent developments, with particular focus on the NP operator’s action on the energy space of the domain. I will also present recent work for domains in 3D with conical points featuring rotational symmetry. In this situation, we have been able to describe the spectrum both for boundary data in L^2 and for data in the energy space. In the former case, the essential spectrum consists of the union of countably many self-intersecting curves in the plane, and outside of this set the index may be computed as the winding number with respect to the essential spectrum. In the latter case the essential spectrum consists of a real interval. Based on joint work with Johan Helsing and Mihai Putinar.