The spectrum of the Poincaré operator in an ellipsoid

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...

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Bibliographic Details
Published inJournal of spectral theory
Main Authors Colin de Verdière, Yves, Vidal, Jérémie
Format Journal Article
LanguageEnglish
Published European Mathematical Society 06.06.2025
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ISSN1664-039X
1664-0403
DOI10.4171/jst/553

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Summary: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 and Rieutord, Phys. Rev. E 95 (2017), article no. 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.
ISSN:1664-039X
1664-0403
DOI:10.4171/jst/553