On the Eigenvalues of the Electrostatic Integral Operator, II.

A fundamental result in scattering and potential theory in R3 states that the spectrum of the electrostatic integral operator lies in the interval [−1,1]. In the case of a sphere and a prolate spheroid, it is known that the spectrum of the operator lies in the interval [−1,0]. A fundamental question arises whether the spectrum of the operator always lies in this interval, or whether there exists a smooth surface for which the electrostatic integral operator has a positive eigenvalue. In this paper, this question is answered. A surface is produced whereby the underlying integral operator has a positive eigenvalue.