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Tr. Mat. Inst. Steklova, 2011, Volume 273, Pages 30–40 (Mi tm3285)  

This article is cited in 3 scientific papers (total in 3 papers)

Topological properties of eigenoscillations in mathematical physics

V. I. Arnold


Abstract: Courant proved that the zeros of the $n$th eigenfunction of the Laplace operator on a compact manifold $M$ divide this manifold into at most $n$ parts. He conjectured that a similar statement is also valid for any linear combination of the first $n$ eigenfunctions. However, later it was found out that some corollaries to this generalized statement contradict the results of quantum field theory. Later, explicit counterexamples were constructed by O. Viro. Nevertheless, the one-dimensional version of Courant's theorem is apparently valid; to prove it, I. M. Gel'fand proposed a method based on the ideas of quantum mechanics and the analysis of the actions of permutation groups. This leads to interesting questions of describing the statistical properties of group representations that arise from their action on eigenfunctions of the Laplace operator. The analysis of these questions entails, among other things, problems of singularity theory.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2011, 273, 25–34

Bibliographic databases:

UDC: 517.9
Received in December 2009

Citation: V. I. Arnold, “Topological properties of eigenoscillations in mathematical physics”, Modern problems of mathematics, Collected papers. In honor of the 75th anniversary of the Institute, Tr. Mat. Inst. Steklova, 273, MAIK Nauka/Interperiodica, Moscow, 2011, 30–40; Proc. Steklov Inst. Math., 273 (2011), 25–34

Citation in format AMSBIB
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\paper Topological properties of eigenoscillations in mathematical physics
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\pages 30--40
\publ MAIK Nauka/Interperiodica
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Jain S.R., Samajdar R., “Nodal Portraits of Quantum Billiards: Domains, Lines, and Statistics”, Rev. Mod. Phys., 89:4 (2017), 045005  crossref  mathscinet  isi
    2. Helffer B., Kiwan R., “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. Dittrich J., Kovarik H., Laptev A., Eur. Math. Soc., 2017, 353–371  mathscinet  zmath  isi
    3. Berard P., Helffer B., “On Courant'S Nodal Domain Property For Linear Combinations of Eigenfunctions. Part i”, Doc. Math., 23 (2018), 1561–1585  isi
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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