The Laplace–Beltrami Operator on the Surface of the Ellipsoid

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.