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Spectral sum rules on a $d$--sphere

Paolo Amore

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 $Σ(Ω) = 1 + κY_{1,\vec{0}}(Ω)$ 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.

Spectral sum rules on a $d$--sphere

Abstract

We derive spectral sum rules for inverse powers of the eigenvalues of the Helmholtz equation on a -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 in 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.

Paper Structure

This paper contains 8 sections, 77 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. General properties
  3. Renormalization
  4. An application
  5. Conclusions
  6. Green's functions
  7. Perturbation theory for the zero mode
  8. Some integrals

Figures (5)

  • Figure 1: Comparison between the numerical eigenvalues on the $3$--sphere with $\ell_{\rm max}=30$ and $\kappa=2$ and the Weyl-law approximation.
  • Figure 2: Difference between the exact sum rule for $d=3$ and $p=3$ and the numerical approximation obtained by using the Rayleigh--Ritz method with $\ell_{\rm max}=30$ and Weyl's law.
  • Figure 3: Comparison between the numerical eigenvalues on the $5$--sphere with $\ell_{\rm max}=15$ and $\kappa=2$ and the Weyl-law approximation.
  • Figure 4: Difference between the exact sum rule for $d=5$ and $p=3$ and the numerical approximation obtained by using the Rayleigh--Ritz method with $\ell_{\rm max}=15$ and Weyl's law.
  • Figure 5: Behavior of $\Delta(5,\ell_{\rm max},3)$ as a function of $\ell_{\rm max}$, where $\Delta(d,\ell_{\rm max},s)$ measures the difference between the exact tail contribution and its Weyl-law approximation (here for $\kappa=0$).