The Laplace-Beltrami Operator on the Surface of the Ellipsoid

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.

References in corpus (3)

• M-dissipative boundary conditions and boundary tuples for Maxwell operators
• Semiclassical analysis of a nonlocal boundary value problem related to magnitude
• On the eigenvalues of the Laplacian on ellipsoids obtained as perturbation of unit sphere