On the Magnetostatic Integral Operator for Ellipsoids

The magnetostatic integral operator in R3 is of considerable interest in both scattering and potential theory. As well as for its scalar pendant, the electrostatic integral operator, its eigenvalues, and its eigenfunctions play a fundamental role in the theory of permanent magnetization. While the eigenvalues of both operators are closely related, explicit expressions for the eigenfunctions of the magnetostatic operator are known only in the case when the underlying surface is a sphere. For the cases of ellipsoids and spheroids, we present explicit expressions for the eigenvalues in terms of Lam´e functions and associated Legendre functions and expressions for eigenfunctions in terms of surface ellipsoidal and surface spheroidal harmonics, respectively. As an application the permanent magnetization of ellipsoids is discussed.