These notes present a partial survey of our recent contributions to the understanding of nodal sets of eigenfunctions (constructions of families of eigenfunctions with few or many nodal domains, equality cases in Courant’s nodal domain theorem), revisiting Antonie Stern’s thesis, Göttingen, 1924.
@article{TSG_2014-2015__32__1_0, author = {Pierre B\'erard and Bernard Helffer}, title = {Nodal sets of eigenfunctions, {Antonie} {Stern{\textquoteright}s} results revisited}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {1--37}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {32}, year = {2014-2015}, doi = {10.5802/tsg.302}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.302/} }
TY - JOUR AU - Pierre Bérard AU - Bernard Helffer TI - Nodal sets of eigenfunctions, Antonie Stern’s results revisited JO - Séminaire de théorie spectrale et géométrie PY - 2014-2015 SP - 1 EP - 37 VL - 32 PB - Institut Fourier PP - Grenoble UR - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.302/ DO - 10.5802/tsg.302 LA - en ID - TSG_2014-2015__32__1_0 ER -
%0 Journal Article %A Pierre Bérard %A Bernard Helffer %T Nodal sets of eigenfunctions, Antonie Stern’s results revisited %J Séminaire de théorie spectrale et géométrie %D 2014-2015 %P 1-37 %V 32 %I Institut Fourier %C Grenoble %U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.302/ %R 10.5802/tsg.302 %G en %F TSG_2014-2015__32__1_0
Pierre Bérard; Bernard Helffer. Nodal sets of eigenfunctions, Antonie Stern’s results revisited. Séminaire de théorie spectrale et géométrie, Volume 32 (2014-2015), pp. 1-37. doi : 10.5802/tsg.302. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.302/
[1] V.I Arnold Topological properties of eigenoscillations in mathematical physics, Proceedings of the Steklov Institute of Mathematics, Volume 273 (2011), pp. 25-34 | MR | Zbl
[2] R. Band; M. Bersudsky; D. Fajman A note on Courant sharp eigenvalues of the Neumann right-angled isosceles triangle (2015) (https://arxiv.org/abs/1507.03410v1)
[3] R. Band; M. Bersudsky; D. Fajman Courant-sharp eigenvalues of Neumann -rep-tiles (2016) (https://arxiv.org/abs/1507.03410v2) | MR
[4] P. Bérard; B. Helffer Partial edited extracts from Antonie Stern’s thesis, Séminaire de Théorie Spectrale et Géométrie, Volume 32, Institut Fourier, 2014-2015
[5] P. Bérard; B. Helffer Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle (2015) (https://arxiv.org/abs/1503.00117, To appear in Letters in Mathematical Physics)
[6] P. Bérard; B. 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
[7] P. Bérard; B. Helffer On the nodal patterns of the 2D isotropic quantum harmonic oscillator (2015) (https://arxiv.org/abs/1506.02374)
[8] P. Bérard; B. 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
[9] L. Bérard Bergery; J.P. Bourguignon Laplacians and submersions with totally geodesic fibers, Illinois Journal of Mathematics, Volume 26 (1982), pp. 181-200 | MR | Zbl
[10] V. Bonnaillie-Noël; B. Helffer Nodal and spectral minimal partitions, The state of the art in 2015 (2015) https://arxiv.org/abs/1506.07249, To appear in the book “Shape optimization and spectral theory”. A. Henrot Ed. (De Gruyter Open)
[11] P. Charron On Pleijel’s theorem for the isotropic harmonic oscillator, Université de Montréal (2015) (Masters thesis)
[12] P. Charron A Pleijel type theorem for the quantum harmonic oscillator (2015) (https://arxiv.org/abs/1512.07880, To appear in J. Spectral Theory)
[13] P. Charron; B. Helffer; T. Hoffmann-Ostenhof Pleijel’s theorem for Schrödinger operators with radial potentials (2016) (https://arxiv.org/abs/1604.08372)
[14] S.Y Cheng Eigenfunctions and nodal sets, Commentarii Mathematici Helvetici, Volume 51 (1976), pp. 43-55 | MR | Zbl
[15] R. Courant Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke, Nachr. Ges. Göttingen (1923), pp. 81-84
[16] R. Courant; D. Hilbert Methods of Mathematical Physics, 1, Wiley-VCH Verlag GmbH & Co. KGaA. New York, 1953 | Zbl
[17] R. Courant; D. Hilbert Methoden der Mathematischen Physik, Heidelberger Taschenbücher Band 30, I, Springer, 1968 (Dritte Auflage) | MR | Zbl
[18] A. Eremenko; D. Jakobson; N. Nadirashvili On nodal sets and nodal domains on , Annales Institut Fourier, Volume 57 (2007), pp. 2345-2360 | Numdam | MR | Zbl
[19] G. Gauthier-Shalom; K. Przybytkowski Description of a nodal set on (2006) Research report (unpublished)
[20] B. Helffer; T. Hoffmann-Ostenhof Minimal partitions for anisotropic tori, Journal of Spectral Theory, Volume 4 (2014), pp. 221-233 | MR
[21] B. Helffer; M. Persson Sundqvist Nodal domains in the square – The Neumann case, Moscow Mathematical Journal, Volume 15 (2015), pp. 455-495 | MR
[22] B. Helffer; M. Persson Sundqvist On nodal domains in Euclidean balls, Proceeding of the American Mathematical Society, Volume 144 (2016), pp. 4777-4791 | MR
[23] D. Jakobson; N. Nadirashvili Eigenvalues with few critical points, Journal of Differential Geometry, Volume 53 (1999), pp. 177-182 | MR | Zbl
[24] N. Kuznetsov On delusive nodal sets of free oscillations, European Mathematical Society Newsletter, Volume 96 (2015), pp. 34-40 | MR
[25] C. Léna Courant-sharp eigenvalues of a two-dimensional torus, C. R. Math. Acad. Sci. Paris, Volume 353 (2015) no. 6, pp. 535-539 (doi:10.1016/j.crma.2015.03.014) | MR
[26] C. Léna On the parity of the number of nodal domains for an eigenfunction of the Laplacian on tori (2015) (https://arxiv.org/abs/1504.03944)
[27] C. Léna Pleijel’s nodal domain theorem for Neumann eigenfunctions (2016) (https://arxiv.org/abs/1609.02331)
[28] H. Lewy On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere, Communications in Partial Differential Equations, Volume 12 (1977), pp. 1233-1244 | MR | Zbl
[29] J. Leydold Knotenlinien und Knotengebiete von Eigenfunktionen, Universität Wien (1989) Diplom Arbeit (unpublished)
[30] J. Leydold On the number of nodal domains of spherical harmonics, Topology, Volume 35 (1996), pp. 301-321 | MR | Zbl
[31] Å. Pleijel Remarks on Courant’s nodal theorem, Communications in Pure and Applied Mathematics, Volume 9 (1956), pp. 543-550 | MR | Zbl
[32] F. Pockels Über die partielle Differentialgleichung und deren Auftreten in mathematischen Physik, Teubner- Leipzig, 1891 (Historical Math. Monographs. Cornell University http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=00880001)
[33] A. Stern Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen, Druck der Dieterichschen Universitäts-Buchdruckerei (W. Fr. Kaestner), Göttingen, Germany (1925) (Ph. D. Thesis)
[34] C. Sturm Mémoire sur les équations différentielles linéaires du second ordre, Journal de Mathématiques Pures et Appliquées, Volume 1 (1836), p. 106-186, 269-277, 375-444 | EuDML
[35] A. Vogt Wissenschaftlerinnen in Kaiser-Wilhelm-Instituten. A-Z, Veröffentlichungen aus dem Archiv der Max-Planck-Gesellschaft, 12, Archiv der Max-Planck-Gesellschaft, 2008
Cited by Sources: