A spectral property of the electrostatic integral operator
Integral operators in general and of the electrostatic type in particular, play a prominent role in mathematical analyses and applications. The author here is concerned with the eigenvalue problems for the electrostatic and adjoint electrostatic integral operators: \[ T,T': C(E)\to C(E) \] defined by means of appropriate integral equations. After stating and proving a preparatory lemma on spherical harmonics, a conjecture on the sum of eigenvalues belonging to \(2n+1\) linearly independent eigenfunctions (lief) is given and discussed. The spectral property is examined at length. It is shown that the arithmetic means of the eigenvalues corresponding to the maximum number of liefs of equal order is the same for the ellipsoid as for the sphere. The work can serve as a practical tool for the determination of ellipsoidal harmonics and issues thereof.
scientific article; zbMATH DE number 1375015