New Spectral Results for the Electrostatic Integral Operator

In this paper, two new fundamental results are established concerning the spectra of both the double-layer potential and the electrostatic integral operators in R 3, While it is known that 0 lies in the spectrum of each operator, its specific nature is not well understood. For some geometries (e.g., a sphere and a prolate spheroid), it is known that 0 lies in their continuous spectra. In this paper, however, specific examples are produced, where 0 lies in the point spectrum of each operator. Thus it is established that the spectral nature of 0 depends upon the geometry of the underlying surface. The second new result given is an important generalization of a recent paper concerning the possible refinement of a classical result of Plemelj.