Topological properties of eigenoscillations in mathematical physics
type: Item
instance of: scholarly article
title: Topological properties of eigenoscillations in mathematical physics
zbMATH Open document ID: 1229.35220
DOI: 10.1134/S0081543811040031
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2011-10-25T00:00:00Z
review text: The author disputes the Courant' results regarding the zeros of the eigenfunctions of Laplace operator. According this result, the zeros of the \(n\)-th eigenfunction of Laplace operator on a compact manifold \(M\) divide this manifold into at most \(n\) parts. As a consequence, a similar result is also valid for any linear combination of the first \(n\) eigenfunctions. But this statement contradicts the results of quantum theory filed. In this sense, O. Viro has constructed an explicit counterexample. Then, in the one-dimensional version of Courant's theorem, Gelfand proposed a method based on the ideas of quantum mechanics and the analysis of the actions of permutation groups, to show that the Corollary of the Courant's result is valid. The work of Gelfand leads to interesting questions of describing the statistical properties of group representations that arise from their action on eigenfunctions of the Laplace operator.
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 58H10
Mathematics Subject Classification ID: 81R10
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 5963528
zbMATH Keywords: eigenoscillations
zbMATH Keywords: eigenfunctions
zbMATH Keywords: Courant's theorem
zbMATH Keywords: quantum mechanics
zbMATH Keywords: singularity theory
author: Vładimir I. Arnold
reviewed by: Marin I. Marin
MaRDI profile type: Publication
cites work: Q5820676
cites work: Structure of representations generated by vectors of highest weight
cites work: Q3309368
cites work: Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect
cites work: On the topology of the eigenfields
cites work: Q4818701
full work available at URL: s0081543811040031
OpenAlex ID: W2011089961
description: scientific article; zbMATH DE number 5963528