The spectrum of the Poincaré operator in an ellipsoid

• Published in: Journal of spectral theory
• Main Authors: Colin de Verdière, Yves, Vidal, Jérémie
• Format: Journal Article
• Language: English
• Published: European Mathematical Society 06.06.2025
• Subjects: Analysis of PDEs; Classical Analysis and ODEs; Differential Geometry; Fluid mechanics; Mathematics; Mechanics; Physics; Spectral Theory; Poincaré equation; propagation of singularities; Weyl asymptotics; boundary pseudo-differential calculus; inertial waves; orthogonal polynomials
• ISSN: 1664-039X; 1664-0403; 1664-0403
• DOI: 10.4171/jst/553

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.