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Using the spherical harmonic functions we understand the eigenvalues and eigenvectors of the Laplacian on the round 2-sphere also the general n-sphere. (1) contains explicit descriptions of the eigenvalues and eigenvectors of the standard basic manifolds including the n-sphere. Of historical interest is the treatment in what is arguably the first textbook on physics by Tait and Thomson. The latter (a.k.a. Lord Kelvin) used it to estimate the age of the sun. Inaccurately, but not due to errors in the mathematics, thermonuclear reactions hadn’t yet been discovered.[MO]

References:

1. Berger, Marcel; Gauduchon, Paul; Mazet, Edmond Le spectre d'une variété riemannienne. Lecture Notes in Mathematics, Vol. 194 Springer-Verlag, Berlin-New York 1971 vii+251 pp.
2. Piotr Hajlasz – Functional Analysis notes. https://sites.pitt.edu/~hajlasz/Notatki/Functional%20Analysis2.pdf