Abstract/Summary

The study of the spectral properties of those operators which output potential functions has historically been of great value, particularly in the resolution of certain problems in potential theory and the mathematical study of gravitational and electromagnetic fields. The initial inspiration for this work was to build on the existing body of results that describe the spectrum of the Neumann Poincar´e operator on different surfaces, particularly those with corners and edges. The particular spectral properties of this integral operator align nicely with a number of scenarios discussed in physics, most notably in the study of plasmonics, where there is a noted coincidence between the elements of the spectrum of the Neumann Poincar´e operator when acting on specific function spaces and the phenomena of plasmon resonances. This work is a study of the spectral properties of the Neumann Poincar´e operator when considered over sets bounded by bow-tie curves, formed of two tear drop shapes or ’wings’, each with a corner that coincides, a scenario that distinguishes itself from prior studies in that the surface being acted on is neither completely smooth nor can it be characterized as a Lipschitz domain in the region of the curve’s singular point.