Nodal sets of eigenfunctions, Antonie Stern’s results revisited

Bérard, Pierre; Helffer, Bernard

Séminaire de théorie spectrale et géométrie, Tome 32 (2014-2015), pp. 1-37.

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.

Classification : 35P15, 49R50
Mots clés : Nodal domains, Courant theorem, Pleijel theorem, Dirichlet Laplacian

[1] Arnold, V.I Topological properties of eigenoscillations in mathematical physics, Proceedings of the Steklov Institute of Mathematics, Volume 273 (2011), pp. 25-34 | MR | Zbl

[2] Band, R.; Bersudsky, M.; Fajman, D. A note on Courant sharp eigenvalues of the Neumann right-angled isosceles triangle (2015) (https://arxiv.org/abs/1507.03410v1)

[3] Band, R.; Bersudsky, M.; Fajman, D. Courant-sharp eigenvalues of Neumann $2$-rep-tiles (2016) (https://arxiv.org/abs/1507.03410v2) | MR

[4] Bérard, P.; Helffer, B. Partial edited extracts from Antonie Stern’s thesis, Séminaire de Théorie Spectrale et Géométrie, Volume 32, Institut Fourier, 2014-2015

[5] Bérard, P.; Helffer, B. 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] Bérard, P.; Helffer, B.; 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 Proceedings in Mathematics & Statistics), Volume 127, Springer International Publishing (2015), pp. 69-114 | MR

[7] Bérard, P.; Helffer, B. On the nodal patterns of the 2D isotropic quantum harmonic oscillator (2015) (https://arxiv.org/abs/1506.02374)

[8] Bérard, P.; Helffer, B. 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] Bérard Bergery, L.; Bourguignon, J.P. Laplacians and submersions with totally geodesic fibers, Illinois Journal of Mathematics, Volume 26 (1982), pp. 181-200 | MR | Zbl

[10] Bonnaillie-Noël, V.; Helffer, B. 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] Charron, P. On Pleijel’s theorem for the isotropic harmonic oscillator, Université de Montréal (2015) (Masters thesis)

[12] Charron, P. A Pleijel type theorem for the quantum harmonic oscillator (2015) (https://arxiv.org/abs/1512.07880, To appear in J. Spectral Theory)

[13] Charron, P.; Helffer, B.; Hoffmann-Ostenhof, T. Pleijel’s theorem for Schrödinger operators with radial potentials (2016) (https://arxiv.org/abs/1604.08372)

[14] Cheng, S.Y Eigenfunctions and nodal sets, Commentarii Mathematici Helvetici, Volume 51 (1976), pp. 43-55 | MR | Zbl

[15] Courant, R. Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke, Nachr. Ges. Göttingen (1923), pp. 81-84

[16] Courant, R.; Hilbert, D. Methods of Mathematical Physics, 1, Wiley-VCH Verlag GmbH & Co. KGaA. New York, 1953 | Zbl

[17] Courant, R.; Hilbert, D. Methoden der Mathematischen Physik, Heidelberger Taschenbücher Band 30, I, Springer, 1968 (Dritte Auflage) | MR | Zbl

[18] Eremenko, A.; Jakobson, D.; Nadirashvili, N. On nodal sets and nodal domains on ${𝕊}^2$, Annales Institut Fourier, Volume 57 (2007), pp. 2345-2360 | Numdam | MR | Zbl

[19] Gauthier-Shalom, G.; Przybytkowski, K. Description of a nodal set on ${𝕋}^2$ (2006) Research report (unpublished)

[20] Helffer, B.; Hoffmann-Ostenhof, T. Minimal partitions for anisotropic tori, Journal of Spectral Theory, Volume 4 (2014), pp. 221-233 | MR

[21] Helffer, B.; Persson Sundqvist, M. Nodal domains in the square – The Neumann case, Moscow Mathematical Journal, Volume 15 (2015), pp. 455-495 | MR

[22] Helffer, B.; Persson Sundqvist, M. On nodal domains in Euclidean balls, Proceeding of the American Mathematical Society, Volume 144 (2016), pp. 4777-4791 | MR

[23] Jakobson, D.; Nadirashvili, N. Eigenvalues with few critical points, Journal of Differential Geometry, Volume 53 (1999), pp. 177-182 | MR | Zbl

[24] Kuznetsov, N. On delusive nodal sets of free oscillations, European Mathematical Society Newsletter, Volume 96 (2015), pp. 34-40 | MR

[25] Léna, C. 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] Léna, C. 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] Léna, C. Pleijel’s nodal domain theorem for Neumann eigenfunctions (2016) (https://arxiv.org/abs/1609.02331)

[28] Lewy, H. 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] Leydold, J. Knotenlinien und Knotengebiete von Eigenfunktionen, Universität Wien (1989) Diplom Arbeit (unpublished) http://othes.univie.ac.at/34443/

[30] Leydold, J. On the number of nodal domains of spherical harmonics, Topology, Volume 35 (1996), pp. 301-321 | MR | Zbl

[31] Pleijel, Å. Remarks on Courant’s nodal theorem, Communications in Pure and Applied Mathematics, Volume 9 (1956), pp. 543-550 | MR | Zbl

[32] Pockels, F. 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] Stern, A. Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen, Druck der Dieterichschen Universitäts-Buchdruckerei (W. Fr. Kaestner), Göttingen, Germany (1925) (Ph. D. Thesis)

[34] Sturm, C. 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 | Numdam

[35] Vogt, A. Wissenschaftlerinnen in Kaiser-Wilhelm-Instituten. A-Z, Veröffentlichungen aus dem Archiv der Max-Planck-Gesellschaft, 12, Archiv der Max-Planck-Gesellschaft, 2008