Topological properties of eigenoscillations in mathematical physics

V. I. Arnold

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, Tome 273 (2011), pp. 30-40

Résumé

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.