Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 067, 21 pages      arXiv:2312.01620      https://doi.org/10.3842/SIGMA.2024.067

The Laplace-Beltrami Operator on the Surface of the Ellipsoid

Hans Volkmer
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, USA

Received December 04, 2023, in final form July 10, 2024; Published online July 25, 2024

Abstract
The Laplace-Beltrami operator on (the surface of) a triaxial ellipsoid admits a sequence of real eigenvalues diverging to plus infinity. By introducing ellipsoidal coordinates, this eigenvalue problem for a partial differential operator is reduced to a two-parameter regular Sturm-Liouville problem involving ordinary differential operators. This two-parameter eigenvalue problem has two families of eigencurves whose intersection points determine the eigenvalues of the Laplace-Beltrami operator. Eigenvalues are approximated numerically through eigenvalues of generalized matrix eigenvalue problems. Ellipsoids close to spheres are studied employing Lamé polynomials.

Key words: Laplace-Beltrami operator; triaxial ellipsoid; two-parameter Sturm-Liouville problem; generalized matrix eigenvalue problem; eigencurves.

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