Edited extracts from Antonie Stern’s thesis
Séminaire de théorie spectrale et géométrie, Volume 32 (2014-2015), pp. 39-72.

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
Bérard, Pierre 1; Helffer, Bernard 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)
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Bérard, Pierre; Helffer, Bernard. Edited extracts from Antonie Stern’s thesis. Séminaire de théorie spectrale et géométrie, Volume 32 (2014-2015), pp. 39-72. doi : 10.5802/tsg.303. http://www.numdam.org/articles/10.5802/tsg.303/

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

[2] Bérard, Pierre; Helffer, Bernard; Baklouti, Ali; El Kacimi, Aziz; Kallel, Sadok; Mir, Nordine 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, Springer International Publishing (2015), pp. 69-114 | MR

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

[5] Vogt, Annette 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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