We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on . We also construct a solution of the equation in that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.
On étudie les configurations topologiques possibles d’ensembles nodaux, en particulier, le nombre de leurs composantes, pour les harmoniques sphériques sur . Nous construisons aussi une solution de l’équation dans qui possède seulement deux domaines nodaux. Cette équation est considérée dans l’étude des fonctions propres à haute énergie.
Keywords: Laplacian, nodal sets, nodal domains, spherical harmonic, topological configuration
Mot clés : Laplacien, ensemble nodaux, domaines nodaux, harmonique sphérique, configuration topologique
Eremenko, Alexandre 1; Jakobson, Dmitry 2; Nadirashvili, Nikolai 3
@article{AIF_2007__57_7_2345_0, author = {Eremenko, Alexandre and Jakobson, Dmitry and Nadirashvili, Nikolai}, title = {On nodal sets and nodal domains on $\mathbf{S^2}$ and ${\mathbb{R}}^{\mathbf{2}}$}, journal = {Annales de l'Institut Fourier}, pages = {2345--2360}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {7}, year = {2007}, doi = {10.5802/aif.2335}, mrnumber = {2394544}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2335/} }
TY - JOUR AU - Eremenko, Alexandre AU - Jakobson, Dmitry AU - Nadirashvili, Nikolai TI - On nodal sets and nodal domains on $\mathbf{S^2}$ and ${\mathbb{R}}^{\mathbf{2}}$ JO - Annales de l'Institut Fourier PY - 2007 SP - 2345 EP - 2360 VL - 57 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2335/ DO - 10.5802/aif.2335 LA - en ID - AIF_2007__57_7_2345_0 ER -
%0 Journal Article %A Eremenko, Alexandre %A Jakobson, Dmitry %A Nadirashvili, Nikolai %T On nodal sets and nodal domains on $\mathbf{S^2}$ and ${\mathbb{R}}^{\mathbf{2}}$ %J Annales de l'Institut Fourier %D 2007 %P 2345-2360 %V 57 %N 7 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2335/ %R 10.5802/aif.2335 %G en %F AIF_2007__57_7_2345_0
Eremenko, Alexandre; Jakobson, Dmitry; Nadirashvili, Nikolai. On nodal sets and nodal domains on $\mathbf{S^2}$ and ${\mathbb{R}}^{\mathbf{2}}$. Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2345-2360. doi : 10.5802/aif.2335. https://aif.centre-mersenne.org/articles/10.5802/aif.2335/
[1] Some unsolved problems in the theory of differential equations and mathematical physics, Uspekhi Mat. Nauk, Volume 44 (1989), pp. 191-202 transl. in Russian Math. Surveys 44 (1989), p.157–171 | MR | Zbl
[2] Methods of Mathematical Physics, I, Interscience Publishers, New York, 1953 | MR | Zbl
[3] Rational functions with real critical points and the B. and M.Shapiro Conjecture in real algebraic geometry, Annals of Math., Volume 155 (2002), pp. 105-129 | DOI | MR | Zbl
[4] The topology of real projective algebraic varieties, Uspehi Mat. Nauk, Volume 178 (1974), pp. 3-79 | MR | Zbl
[5] Geometric properties of eigenfunctions, Russian Math. Surveys, Volume 56 (2001), pp. 67-88 | DOI | MR | Zbl
[6] Topology of zeros of eigenfunctions, Funct. Anal. Appl., Volume 23 (1989), pp. 218-220 | DOI | MR | Zbl
[7] The number of components of the complement of the level surface of a harmonic polynomial in three variables, Funct. Anal. Appl., Volume 28 (1994), pp. 116-118 | DOI | MR | Zbl
[8] On the number of components of the complement to some algebraic curves, Russian Math. Surveys, Volume 57 (2002), pp. 1228-1229 | DOI | MR | Zbl
[9] Graphs on surfaces and their applications. With an appendix by Don B. Zagier, Low-Dimensional Topology, II, Encyclopaedia of Mathematical Sciences, Volume 141, Springer-Verlag, Berlin, 2004 | MR | Zbl
[10] On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere, Comm. PDE,, Volume 2 (1977), pp. 1233-1244 | DOI | MR | Zbl
[11] On the number of nodal domains of spherical harmonics, Topology, Volume 35 (1996), pp. 301-321 | DOI | MR | Zbl
[12] Asymptotic distribution of nodal sets on spheres, Johns Hopkins University, Baltimore, MD 2000 (1994) (Ph. D. Thesis)
[13] Remarks on Courant’s nodal line theorem, Comm. Pure Appl. Math, Volume 9 (1956), pp. 543-550 | DOI | Zbl
[14] Optimal degree construction of real algebraic plane nodal curves with prescribed topology. I. The orientable case. Real algebraic and analytic geometry (Segovia, 1995), Rev. Mat. Univ. Complut. Madrid, Volume 10 (1997), pp. 291-310 (Special Issue suppl.) | MR | Zbl
[15] Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros, Math. Comp., Volume 70 (2001), pp. 1205-1220 | DOI | MR | Zbl
[16] Equations of Mathematical Physics, Moscow, 1953 | Zbl
[17] Real algebraic plane curves: constructions with controlled topology, Leningrad Math. J., Volume 1 (1990), pp. 1059-1134 | MR | Zbl
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