Spectral sum rules on a $d$--sphere
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914437321981952 |
|---|---|
| author | Amore, Paolo |
| author_facet | Amore, Paolo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00845 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral sum rules on a $d$--sphere Amore, Paolo Mathematical Physics 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. |
| title | Spectral sum rules on a $d$--sphere |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2604.00845 |