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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2011, Volume 273, Pages 30–40 (Mi tm3285)  

This article is cited in 10 scientific papers (total in 10 papers)

Topological properties of eigenoscillations in mathematical physics

V. I. Arnold
References:
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.
Received in December 2009
English version:
Proceedings of the Steklov Institute of Mathematics, 2011, Volume 273, Pages 25–34
DOI: https://doi.org/10.1134/S0081543811040031
Bibliographic databases:
Document Type: Article
UDC: 517.9
Language: Russian
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, Trudy 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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\by V.~I.~Arnold
\paper Topological properties of eigenoscillations in mathematical physics
\inbook Modern problems of mathematics
\bookinfo Collected papers. In honor of the 75th anniversary of the Institute
\serial Trudy Mat. Inst. Steklova
\yr 2011
\vol 273
\pages 30--40
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\transl
\jour Proc. Steklov Inst. Math.
\yr 2011
\vol 273
\pages 25--34
\crossref{https://doi.org/10.1134/S0081543811040031}
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  • This publication is cited in the following 10 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
     
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