Edited extracts from Antonie Stern’s thesis
Séminaire de théorie spectrale et géométrie, Volume 32  (2014-2015), p. 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 : https://doi.org/10.5802/tsg.303
Classification:  35P15,  49R50
Keywords: Nodal domains, Nodal lines, Courant theorem
     author = {B\'erard, Pierre and Helffer, Bernard},
     title = {Edited extracts from Antonie Stern's thesis},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {32},
     year = {2014-2015},
     pages = {39-72},
     doi = {10.5802/tsg.303},
     language = {en},
     url = {http://www.numdam.org/item/TSG_2014-2015__32__39_0}
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/item/TSG_2014-2015__32__39_0/

[1] Bérard, Pierre; Helffer, Bernard Nodal sets of eigenfunctions, Antonie Stern’s results revisited, Séminaire de Théorie Spectrale et Géométrie, Institut Fourier, Grenoble, Tome 32 (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 International Publishing (Springer Proceedings in Mathematics & Statistics) Tome 127 (2015), pp. 69-114 | MR 3445517

[3] Bérard, Pierre; Helffer, Bernard A. Stern’s analysis of the nodal sets of some families of spherical harmonics revisited, Monatshefte für Mathematik, Tome 180 (2016), pp. 435-468 | Article | MR 3513215

[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, Archiv der Max-Planck-Gesellschaft, Veröffentlichungen aus dem Archiv der Max-Planck-Gesellschaft, Tome 12 (2008)