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.
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{\textquoteright}s} thesis},
journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
pages = {39--72},
publisher = {Institut Fourier},
address = {Grenoble},
volume = {32},
year = {2014-2015},
doi = {10.5802/tsg.303},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/tsg.303/}
}
TY - JOUR AU - Bérard, Pierre AU - Helffer, Bernard TI - Edited extracts from Antonie Stern’s thesis JO - Séminaire de théorie spectrale et géométrie PY - 2014-2015 SP - 39 EP - 72 VL - 32 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/tsg.303/ DO - 10.5802/tsg.303 LA - en ID - TSG_2014-2015__32__39_0 ER -
%0 Journal Article %A Bérard, Pierre %A Helffer, Bernard %T Edited extracts from Antonie Stern’s thesis %J Séminaire de théorie spectrale et géométrie %D 2014-2015 %P 39-72 %V 32 %I Institut Fourier %C Grenoble %U http://archive.numdam.org/articles/10.5802/tsg.303/ %R 10.5802/tsg.303 %G en %F 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://archive.numdam.org/articles/10.5802/tsg.303/
[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, Springer International Publishing (2015), pp. 69-114 | MR
[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 | DOI | MR
[4] Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen, Druck der Dieterichschen Universitäts-Buchdruckerei (W. Fr. Kaestner), Göttingen, Germany (1925) (Ph. D. Thesis)
[5] 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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