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

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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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
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
3. Berard P., Helffer B., “On Courant'S Nodal Domain Property For Linear Combinations of Eigenfunctions. Part i”, Doc. Math., 23 (2018), 1561–1585
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