A. Stern's analysis of the nodal sets of some families of spherical harmonics revisited (Q530563)

The analysis of \textit{A. Stern} [Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen. Math. Naturwiss. Diss. Göttingen (1925; JFM 51.0356.01)] and \textit{H. Lewy} [Commun. Partial Differ. Equations 2, 1233--1244 (1977; Zbl 0377.31008)] devoted to construction of spherical harmonics with two or three nodal domains is revisited. In Section 2 some properties of spherical harmonics and Legendre functions are recalled. In Section 3 a proof of the following Stern's theorem is given: Let \(\mathbb S^2\) be the unit sphere in \(\mathbb R^3\) and \(\Delta\) the non-positive spherical Laplacian then for any odd integer \(l\) there exists a spherical harmonic of degree \(l\) whose nodal set consists of a single simple closed curve (as a consequence, \(u\) has exactly two nodal domains). In Section 4 Stern's theorem concerning the case of even \(l\) is proved: Let \(\mathbb S^2\) be the unit sphere in \(\mathbb R^3\) then for any even integer \(l\geq2\) there exists a spherical harmonic of degree \(l\) whose nodal set consists of two disjoint simple closed curves (as a consequence, \(u\) has exactly three nodal domains). In Section 5 the Courant sharp property and open questions for minimal partitions for the sphere are discussed. In Appendix (Section 6) some numerical computations of nodal sets of spherical harmonics are given.