A Sum-Property of the Eigenvalues of the Electrostatic Integral Operator

The double-layer potential integral operator and the electrostatic integral operator in R3 occur while using integral equation methods in scattering and potential theory. If the underlying surface is either a sphere, a spheroid, or a triaxial ellipsoid, explicit expressions of the eigenvalues and eigenfunctions of order n ∈ N0 are known. The main result of this paper is the following: The sum of all eigenvalues of order n ∈ N0, each counted with respect to its multiplicity, is −1. This is trivial for the sphere. In the case of spheroids, a proof is given for n ∈ N0. For the triaxial ellipsoid, this "sum property" is verified for n = 0, 1, 2, 3, whereas for n > 3 some numerical results are provided.