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Mosc. Math. J., 2015, Volume 15, Number 3, Pages 455–495 (Mi mmj571)  

This article is cited in 7 scientific papers (total in 7 papers)

Nodal domains in the square – the Neumann case

Bernard Helfferab, Mikael Persson Sundqvistc

a Laboratoire Jean Leray, Université de Nantes, France
b Laboratoire de Mathématiques UMR CNRS 8628, Université Paris-Sud-Bât 425, F-91405 Orsay Cedex, France
c Lund University, Department of Mathematical Sciences, Lund, Sweden

Abstract: Å. Pleijel has proved that in the case of the Laplacian on the square with Neumann condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues. We identify five Courant sharp eigenvalues for the Neumann Laplacian in the square, and prove that there are no other cases.

Key words and phrases: nodal domains, Courant theorem, square, Neumann.

DOI: https://doi.org/10.17323/1609-4514-2015-15-3-455-495

Full text: http://www.mathjournals.org/.../2015-015-003-004.html
References: PDF file   HTML file

Bibliographic databases:

MSC: 35B05, 35P20, 58J50
Received: November 28, 2014; in revised form March 4, 2015
Language:

Citation: Bernard Helffer, Mikael Persson Sundqvist, “Nodal domains in the square – the Neumann case”, Mosc. Math. J., 15:3 (2015), 455–495

Citation in format AMSBIB
\Bibitem{HelSun15}
\by Bernard~Helffer, Mikael~Persson~Sundqvist
\paper Nodal domains in the square~-- the Neumann case
\jour Mosc. Math.~J.
\yr 2015
\vol 15
\issue 3
\pages 455--495
\mathnet{http://mi.mathnet.ru/mmj571}
\crossref{https://doi.org/10.17323/1609-4514-2015-15-3-455-495}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3427435}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000365392600004}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. P. Berard, B. Helffer, “Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle”, Lett. Math. Phys., 106:12 (2016), 1729–1789  crossref  mathscinet  zmath  isi  scopus
    2. B. Helffer, M. P. Sundqvist, “On nodal domains in Euclidean balls”, Proc. Amer. Math. Soc., 144:11 (2016), 4777–4791  crossref  mathscinet  zmath  isi  scopus
    3. C. Léna, “Courant-sharp eigenvalues of the three-dimensional square torus”, Proc. Amer. Math. Soc., 144:9 (2016), 3949–3958  crossref  mathscinet  zmath  isi  scopus
    4. P. Berard, B. Helffer, “A. Stern's analysis of the nodal sets of some families of spherical harmonics revisited”, Monatsh. Math., 180:3 (2016), 435–468  crossref  mathscinet  zmath  isi  scopus
    5. R. Band, M. Bersudsky, D. Fajman, “Courant-sharp eigenvalues of Neumann 2-rep-tiles”, Lett. Math. Phys., 107:5 (2017), 821–859  crossref  mathscinet  zmath  isi  scopus
    6. S. R. Jain, R. Samajdar, “Nodal portraits of quantum billiards: domains, lines, and statistics”, Rev. Mod. Phys., 89:4 (2017), 045005  crossref  mathscinet  isi  scopus
    7. B. Helffer, R. Kiwan, “Dirichlet eigenfunctions in the cube, sharpening the Courant nodal inequality”, Functional Analysis and Operator Theory For Quantum Physics: the Pavel Exner Anniversary Volume, EMS Ser. Congr. Rep., eds. J. Dittrich, H. Kovarik, A. Laptev, Eur. Math. Soc., 2017, 353–371  mathscinet  zmath  isi
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