# Exact sum rules for heterogeneous spherical drums

## TL;DR

This paper derives explicit integral formulas for the sum of inverse powers of Laplacian eigenvalues on a sphere with variable density, including renormalization to handle divergences, and verifies results numerically.

## Contribution

It provides the first explicit integral expressions for sum rules of Laplacian eigenvalues on heterogeneous spherical domains, including a renormalization approach for divergence removal.

## Key findings

- Derived explicit integral sum rules for variable density spherical drums.
- Applied formulas to specific cases and verified with numerical eigenvalue calculations.
- Extended understanding of spectral properties of heterogeneous spherical domains.

## Abstract

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.