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.
Classification : 35P15, 49R50
Mots clés : Nodal domains, Nodal lines, Courant theorem
@article{TSG_2014-2015__32__39_0, 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}, pages = {39--72}, publisher = {Institut Fourier}, volume = {32}, year = {2014-2015}, doi = {10.5802/tsg.303}, language = {en}, url = {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, Tome 32 (2014-2015) , pp. 39-72. doi : 10.5802/tsg.303. http://www.numdam.org/item/TSG_2014-2015__32__39_0/
[1] Nodal sets of eigenfunctions, Antonie Stern’s results revisited, Séminaire de Théorie Spectrale et Géométrie, Volume 32, Institut Fourier, Grenoble, 2014-2015
[2] 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 3445517
[3] A. Stern’s analysis of the nodal sets of some families of spherical harmonics revisited, Monatshefte für Mathematik, Volume 180 (2016), pp. 435-468 | Article | MR 3513215
[4] Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen (1925) (Ph. D. Thesis)
[5] Wissenschaftlerinnen in Kaiser-Wilhelm-Instituten. A-Z, Veröffentlichungen aus dem Archiv der Max-Planck-Gesellschaft, Volume 12, Archiv der Max-Planck-Gesellschaft, 2008