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

In this paper, we revisit the analyses of Stern (Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen, W. Fr. Kaestner, Göttingen, 1925 ) and Lewy (Commun Partial Differ Equ 2(12):1233–1244, 1977 ) devoted to the construction of spherical harmonics with two or three nodal do...

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Bibliographic Details
Published inMonatshefte für Mathematik Vol. 180; no. 3; pp. 435 - 468
Main Authors Bérard, P., Helffer, B.
Format Journal Article
LanguageEnglish
Published Vienna Springer Vienna 01.07.2016
Springer Verlag [1948-....]
Subjects
Analysis of PDEs
Differential Geometry
Mathematics
Mathematics and Statistics
Spectral Theory
35B05
Nodal domains
Courant theorem
35P20
Nodal lines
58J50
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Summary:In this paper, we revisit the analyses of Stern (Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen, W. Fr. Kaestner, Göttingen, 1925 ) and Lewy (Commun Partial Differ Equ 2(12):1233–1244, 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 .
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-015-0788-6