Exact sum rules for heterogeneous spherical drums

We have obtained explicit integral expressions for the sums of inverse powers of the eigenvalues of the Laplacian on a unit sphere, in presence of an arbitrary variable density. The exact expressions for the sum rules are obtained by properly "renormalizing" the series, excluding the divergent contribution of the vanishing lowest eigenvalue. For a non--trivial example of a variable density we have applied our formulas to calculate the exact sum rules of order two and three, and we have verified these results calculating the sum rules numerically using the eigenvalues obtained with the Rayleigh-Ritz method.