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Séminaire de Théorie spectrale et géométrie (Grenoble)

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Pierre Bérard; Bernard Helffer
Nodal sets of eigenfunctions, Antonie Stern’s results revisited
Séminaire de Théorie spectrale et géométrie (Grenoble), 32 (2014-2015), p. 1-37, doi: 10.5802/tsg.302
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Class. Math.: 35P15, 49R50
Mots clés: Nodal domains, Courant theorem, Pleijel theorem, Dirichlet Laplacian

Résumé - Abstract

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.

Bibliographie

[1] V.I Arnold, “Topological properties of eigenoscillations in mathematical physics”, Proceedings of the Steklov Institute of Mathematics 273 (2011), p. 25-34  MR 2893541 |  Zbl 1229.35220
[2] R. Band, M. Bersudsky & D. Fajman, “A note on Courant sharp eigenvalues of the Neumann right-angled isosceles triangle”, https://arxiv.org/abs/1507.03410v1, 2015
[3] R. Band, M. Bersudsky & D. Fajman, “Courant-sharp eigenvalues of Neumann $2$-rep-tiles”, https://arxiv.org/abs/1507.03410v2, 2016  MR 3535866
[4] P. Bérard & B. Helffer, Partial edited extracts from Antonie Stern’s thesis, Séminaire de Théorie Spectrale et Géométrie, Institut Fourier, 2014-2015
[5] P. Bérard & B. Helffer, “Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle”, https://arxiv.org/abs/1503.00117, To appear in Letters in Mathematical Physics, 2015
[6] P. Bérard & B. Helffer, Dirichlet eigenfunctions of the square membrane: Courant’s property, and A. Stern’s and Å. Pleijel’s analyses, in Ali Baklouti, Aziz El Kacimi, Sadok Kallel, Nordine Mir, éd., Analysis and Geometry. MIMS-GGTM, Tunis, Tunisia, March 2014. In Honour of Mohammed Salah Baouendi, Springer Proceedings in Mathematics & Statistics, Springer International Publishing, 2015, p. 69-114  MR 3445517
[7] P. Bérard & B. Helffer, “On the nodal patterns of the 2D isotropic quantum harmonic oscillator”, https://arxiv.org/abs/1506.02374, 2015
[8] P. Bérard & B. Helffer, “A. Stern’s analysis of the nodal sets of some families of spherical harmonics revisited”, Monatshefte für Mathematik 180 (2016), p. 435-468 Article |  MR 3513215
[9] L. Bérard Bergery & J.P. Bourguignon, “Laplacians and submersions with totally geodesic fibers”, Illinois Journal of Mathematics 26 (1982), p. 181-200  MR 650387 |  Zbl 0483.58021
[10] V. Bonnaillie-Noël & B. Helffer, “Nodal and spectral minimal partitions, The state of the art in 2015”, https://arxiv.org/abs/1506.07249, To appear in the book “Shape optimization and spectral theory”. A. Henrot Ed. (De Gruyter Open), 2015
[11] P. Charron, On Pleijel’s theorem for the isotropic harmonic oscillator, Masters thesis, Université de Montréal, 2015
[12] P. Charron, “A Pleijel type theorem for the quantum harmonic oscillator”, https://arxiv.org/abs/1512.07880, To appear in J. Spectral Theory, 2015
[13] P. Charron, B. Helffer & T. Hoffmann-Ostenhof, “Pleijel’s theorem for Schrödinger operators with radial potentials”, https://arxiv.org/abs/1604.08372, 2016
[14] S.Y Cheng, “Eigenfunctions and nodal sets”, Commentarii Mathematici Helvetici 51 (1976), p. 43-55  MR 397805 |  Zbl 0334.35022
[15] R. Courant, “Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke”, Nachr. Ges. Göttingen (1923), p. 81-84  JFM 49.0342.01
[16] R. Courant & D. Hilbert, Methods of Mathematical Physics 1, Wiley-VCH Verlag GmbH & Co. KGaA. New York, 1953  Zbl 0053.02805
[17] R. Courant & D. Hilbert, Methoden der Mathematischen Physik, Heidelberger Taschenbücher Band 30 I, Springer, 1968, Dritte Auflage  MR 344038 |  Zbl 0156.23201
[18] A. Eremenko, D. Jakobson & N. Nadirashvili, “On nodal sets and nodal domains on $\mathbb{S}^2$”, Annales Institut Fourier 57 (2007), p. 2345-2360  MR 2394544 |  Zbl 1178.58012
[19] G. Gauthier-Shalom & K. Przybytkowski, Description of a nodal set on $\mathbb{T}^2$, Technical report, McGill University, 2006
[20] B. Helffer & T. Hoffmann-Ostenhof, “Minimal partitions for anisotropic tori”, Journal of Spectral Theory 4 (2014), p. 221-233  MR 3232810
[21] B. Helffer & M. Persson Sundqvist, “Nodal domains in the square – The Neumann case”, Moscow Mathematical Journal 15 (2015), p. 455-495  MR 3427435
[22] B. Helffer & M. Persson Sundqvist, “On nodal domains in Euclidean balls”, Proceeding of the American Mathematical Society 144 (2016), p. 4777-4791  MR 3544529
[23] D. Jakobson & N. Nadirashvili, “Eigenvalues with few critical points”, Journal of Differential Geometry 53 (1999), p. 177-182  MR 1776094 |  Zbl 1038.58036
[24] N. Kuznetsov, “On delusive nodal sets of free oscillations”, European Mathematical Society Newsletter 96 (2015), p. 34-40  MR 3379499
[25] C. Léna, “Courant-sharp eigenvalues of a two-dimensional torus”, C. R. Math. Acad. Sci. Paris 353 (2015) no. 6, p. 535-539, doi:10.1016/j.crma.2015.03.014  MR 3348988
[26] C. Léna, “On the parity of the number of nodal domains for an eigenfunction of the Laplacian on tori”, https://arxiv.org/abs/1504.03944, 2015
[27] C. Léna, “Pleijel’s nodal domain theorem for Neumann eigenfunctions”, https://arxiv.org/abs/1609.02331, 2016
[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 12 (1977), p. 1233-1244  MR 477199 |  Zbl 0377.31008
[29] J. Leydold, Knotenlinien und Knotengebiete von Eigenfunktionen, Diplom Arbeit (unpublished), Universität Wien, 1989
[30] J. Leydold, “On the number of nodal domains of spherical harmonics”, Topology 35 (1996), p. 301-321  MR 1380499 |  Zbl 0853.33012
[31] Å. Pleijel, “Remarks on Courant’s nodal theorem”, Communications in Pure and Applied Mathematics 9 (1956), p. 543-550  MR 80861 |  Zbl 0070.32604
[32] F. Pockels, Über die partielle Differentialgleichung $\Delta u + k^2 u=0$ 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, Ph. D. Thesis, Druck der Dieterichschen Universitäts-Buchdruckerei (W. Fr. Kaestner), Göttingen, Germany, 1925  JFM 51.0356.01
[34] C. Sturm, “Mémoire sur les équations différentielles linéaires du second ordre”, Journal de Mathématiques Pures et Appliquées 1 (1836), p. 106-186, 269-277, 375-444
[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
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