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

Pierre Bérard (1) , Bernard Helffer (2, 3)

1 IF - Institut Fourier
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 LMJL - Laboratoire de Mathématiques Jean Leray

In this paper, we revisit the analyses of Antonie Stern (1925) and Hans Lewy (1977) devoted to the construction of spherical harmonics with two or three nodal domains. Our method yields sharp quantitative results and a better understanding of the occurrence of bifurcations in the families of nodal sets.This paper is a natural continuation of our critical reading of A. Stern's results for Dirichlet eigenfunctions in the square, see arXiv:14026054.

Mots clés: Courant theorem; Nodal domains; Nodal lines

Domaines: Théorie spectrale [math.SP]; Géométrie différentielle [math.DG]; Equations aux dérivées partielles [math.AP]

Pierre Bérard, Bernard Helffer. A. Stern's analysis of the nodal sets of some families of spherical harmonics revisited. Monatshefte für Mathematik, 2016, 180 (3), pp.435--468. ⟨hal-01026363v4⟩