On nodal sets and nodal domains on S 2 and ℝ 2

Alexandre Eremenko[1]; Dmitry Jakobson[2]; Nikolai Nadirashvili[3]

  • [1] Purdue University Mathematics Department 150 N University Street West Lafayette, IN 47907-2067 (USA)
  • [2] McGill University Department of Mathematics and Statistics 805 Sherbrooke Str.West Montreal, QC H3A 2K6 (Canada)
  • [3] Université de Provence Laboratoire d’Analyse, Topologie, Probabilités UMR 6632 Centre de Mathématiques et Informatique 39 rue F.Joliot-Curie 13453 Marseille Cedex 13 (France)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 7, page 2345-2360
  • ISSN: 0373-0956

Abstract

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We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on S 2 . We also construct a solution of the equation Δ u = u in ℝ 2 that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.

How to cite

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Eremenko, Alexandre, Jakobson, Dmitry, and Nadirashvili, Nikolai. "On nodal sets and nodal domains on $\mathbf{S^2}$ and ${\mathbb{R}}^{\mathbf{2}}$." Annales de l’institut Fourier 57.7 (2007): 2345-2360. <http://eudml.org/doc/10299>.

@article{Eremenko2007,
abstract = {We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on $S^2$. We also construct a solution of the equation $\Delta u=u$ in $\{\mathbb\{R\}\}^2$ that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.},
affiliation = {Purdue University Mathematics Department 150 N University Street West Lafayette, IN 47907-2067 (USA); McGill University Department of Mathematics and Statistics 805 Sherbrooke Str.West Montreal, QC H3A 2K6 (Canada); Université de Provence Laboratoire d’Analyse, Topologie, Probabilités UMR 6632 Centre de Mathématiques et Informatique 39 rue F.Joliot-Curie 13453 Marseille Cedex 13 (France)},
author = {Eremenko, Alexandre, Jakobson, Dmitry, Nadirashvili, Nikolai},
journal = {Annales de l’institut Fourier},
keywords = {Laplacian; nodal sets; nodal domains; spherical harmonic; topological configuration},
language = {eng},
number = {7},
pages = {2345-2360},
publisher = {Association des Annales de l’institut Fourier},
title = {On nodal sets and nodal domains on $\mathbf\{S^2\}$ and $\{\mathbb\{R\}\}^\{\mathbf\{2\}\}$},
url = {http://eudml.org/doc/10299},
volume = {57},
year = {2007},
}

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
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 7
SP - 2345
EP - 2360
AB - We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on $S^2$. We also construct a solution of the equation $\Delta u=u$ in ${\mathbb{R}}^2$ that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.
LA - eng
KW - Laplacian; nodal sets; nodal domains; spherical harmonic; topological configuration
UR - http://eudml.org/doc/10299
ER -

References

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  11. J. Leydold, On the number of nodal domains of spherical harmonics, Topology 35 (1996), 301-321 Zbl0853.33012MR1380499
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