Generic properties of the Neumann-Poincaré operator: simplicity of eigenvalues and cyclic vectors

Abstract

Two generic properties of the Neumann--Poincaré operator are investigated. We prove that non-zero eigenvalues of the Neumann--Poincaré operator on smooth boundaries in three dimensions and higher are generically simple in the sense of Baire category. We also prove that the functions defined by the fundamental solutions to the Laplace operator located at points outside the surface are generically cyclic vectors in the sense that the collection of those points where the functions are not cyclic vectors is of measure zero.

15 pages