The spectrum of the Poincar{é} operator in an ellipsoid
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911019802034176 |
|---|---|
| author | de Verdière, Yves Colin Vidal, Jérémie |
| author_facet | de Verdière, Yves Colin Vidal, Jérémie |
| contents | We study the spectrum of the Poincaré operator in triaxial ellipsoids subject to a constant rotation. As explained in the paper, this mathematical problem is interesting for many physical applications. It is known that the spectrum of this bounded self-adjoint operator is pure point with polynomial eigenvectors [Backus & Rieutord, Phys. Rev. E 95 (2017), 053116]. We give two new proofs of this result. Moreover, we describe the large-degree asymptotics of the restriction of that operator to polynomial vector fields of fixed degrees. The main tool is the microlocal analysis of the partial differential equation satisfied by the orthogonal polynomials in ellipsoids. This work also contains numerical calculations of these spectra, showing a very good agreement with the mathematical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01369 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The spectrum of the Poincar{é} operator in an ellipsoid de Verdière, Yves Colin Vidal, Jérémie Analysis of PDEs Mathematical Physics Classical Analysis and ODEs Differential Geometry Spectral Theory 35Q35 (Primary), 53Z05, 76U60, 76B70 (Secondary) We study the spectrum of the Poincaré operator in triaxial ellipsoids subject to a constant rotation. As explained in the paper, this mathematical problem is interesting for many physical applications. It is known that the spectrum of this bounded self-adjoint operator is pure point with polynomial eigenvectors [Backus & Rieutord, Phys. Rev. E 95 (2017), 053116]. We give two new proofs of this result. Moreover, we describe the large-degree asymptotics of the restriction of that operator to polynomial vector fields of fixed degrees. The main tool is the microlocal analysis of the partial differential equation satisfied by the orthogonal polynomials in ellipsoids. This work also contains numerical calculations of these spectra, showing a very good agreement with the mathematical results. |
| title | The spectrum of the Poincar{é} operator in an ellipsoid |
| topic | Analysis of PDEs Mathematical Physics Classical Analysis and ODEs Differential Geometry Spectral Theory 35Q35 (Primary), 53Z05, 76U60, 76B70 (Secondary) |
| url | https://arxiv.org/abs/2305.01369 |