Spectral sum rules on a $d$--sphere

TL;DR

This paper develops spectral sum rules for the eigenvalues of the Helmholtz equation on a d-sphere with arbitrary density, using a renormalization scheme to handle divergences and avoid explicit eigenvalue calculations.

Contribution

It introduces a rigorous renormalization approach to derive exact spectral sum rules on a d-sphere with arbitrary density, bypassing the need for explicit eigenvalues.

Findings

Derived explicit sum rules for specific densities in 3, 4, 5 dimensions.

Validated sum rules with numerical estimates using Rayleigh--Ritz and Weyl's formula.

Removed divergences of the zero mode through a novel renormalization scheme.

Abstract

We derive spectral sum rules for inverse powers of the eigenvalues of the Helmholtz equation on a $d$-sphere in the presence of an arbitrary density. By adopting a rigorous renormalization scheme, we remove the divergent contributions of the zero mode and obtain exact expressions for the sum rules without requiring an explicit determination of the eigenvalues, which is generally impossible. As an application, we derive explicit sum rules for the density $\Sigma(\Omega) = 1 + \kappa Y_{1,\vec{0}}(\Omega)$ in $d=3,4,5$ dimensions and compare them with numerical estimates obtained by approximating the low-lying part of the spectrum with the Rayleigh--Ritz method and the high-energy part with Weyl's formula.