Edited extracts from Antonie Stern’s thesis

Pierre Bérard; Bernard Helffer

doi:10.5802/tsg.303

Pierre Bérard ¹ ; Bernard Helffer ^2,³

¹ Institut Fourier, Université Grenoble Alpes, B.P.74, 38402 Saint-Martin-d’Hères Cedex (France)
² Laboratoire de Mathématiques, Univ. Paris-Sud 11 and CNRS, 91405 Orsay Cedex (France)
³ and Laboratoire de Mathématiques Jean Leray, Université de Nantes, 44322 Nantes (France)

Séminaire de théorie spectrale et géométrie, Tome 32 (2014-2015), pp. 39-72.

Résumé

The main purpose of these notes is to provide a reproduction, with some editing, of the first part of Antonie Stern’s 1924 PhD thesis which deals with the construction of higher energy eigenfunctions with a prescribed number of nodal domains, in contrast with Courant’s nodal domain theorem. A. Stern considers both Dirichlet eigenfunctions for the square membrane – her main result in this framework is mentioned in the second edition of the classical book by R. Courant and D. Hilbert – and eigenfunctions of the spherical Laplacian, her spherical results seem to have been overlooked in the literature, until very recently.

DOI : 10.5802/tsg.303

Classification : 35P15, 49R50
Keywords: Nodal domains, Nodal lines, Courant theorem

Affiliations des auteurs :

Pierre Bérard ¹ ; Bernard Helffer ^{2, 3}

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Pierre Bérard; Bernard Helffer. Edited extracts from Antonie Stern’s thesis. Séminaire de théorie spectrale et géométrie, Tome 32 (2014-2015), pp. 39-72. doi : 10.5802/tsg.303. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.303/

Bibliographie
Cité par

[1] Pierre Bérard; Bernard Helffer Nodal sets of eigenfunctions, Antonie Stern’s results revisited, Séminaire de Théorie Spectrale et Géométrie, Volume 32, Institut Fourier, Grenoble, 2014-2015

[2] Pierre Bérard; Bernard Helffer; Ali Baklouti; Aziz El Kacimi; Sadok Kallel; Nordine Mir Dirichlet eigenfunctions of the square membrane: Courant’s property, and A. Stern’s and Å. Pleijel’s analyses, Analysis and Geometry. MIMS-GGTM, Tunis, Tunisia, March 2014. In Honour of Mohammed Salah Baouendi (Springer Proceedings in Mathematics & Statistics), Volume 127 (2015), pp. 69-114 | MR

[3] Pierre Bérard; Bernard Helffer A. Stern’s analysis of the nodal sets of some families of spherical harmonics revisited, Monatshefte für Mathematik, Volume 180 (2016), pp. 435-468 | DOI | MR

[4] Antonie Stern Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen, Druck der Dieterichschen Universitäts-Buchdruckerei (W. Fr. Kaestner), Göttingen, Germany (1925) (Ph. D. Thesis)

[5] Annette Vogt Wissenschaftlerinnen in Kaiser-Wilhelm-Instituten. A-Z, Veröffentlichungen aus dem Archiv der Max-Planck-Gesellschaft, 12, Archiv der Max-Planck-Gesellschaft, 2008

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