The spectrum of the Poincar{\'e} operator in an ellipsoid

We study the spectrum of the Poincar\'e 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.