Topological properties of eigenoscillations in mathematical physics
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.
scientific article; zbMATH DE number 5963528