On delusive nodal sets of free oscillations (Q2352837)

The author reviews theorems and conjectures about nodal sets of spectral problems. Citing examples, he explains that surprisingly many assertions were incorrect but that later analysis lead to better understanding. The first example relates to Courant's theorem [\textit{R. Courant}, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1923, 81--84 (1923; JFM 49.0342.01)] which says that an eigenfunction of the \(n\)-th Dirichlet eigenvalue decomposes the domain into at most \(n\) nodal domains. (Eigenvalues are sorted in increasing order.) A footnote in the textbook of Courant-Hilbert claims that this bound on nodal domains also holds for arbitrary linear combinations of eigenfunctions corresponding to \(k\)-th eigenvalues with \(k\leq n\). Arnold noticed that this claim cannot hold in general; see \textit{V. I. Arnold} [Proc. Steklov Inst. Math. 273, 25--34 (2011); translation from Tr. Mat. Inst. Steklova 273, 30--40 (2011; Zbl 1229.35220)]. However, for one-dimensional (Sturm-Liouville) eigenvalue problems there is strong evidence given by Gelfand that this claim, named Courant-Gelfand theorem by Arnold, is true, but a formal proof has not yet been written. The second example is the sloshing eigenvalue problem for the free motion of water in a canal. Here, it was falsely asserted in the literature that nodal curves have one end on the free surface and the other on the canal bottom. The third and last example deals with the membrane eigenvalue problem. A conjecture of Payne stated that the second eigenfunction \(u_2\) cannot possess a closed nodal line. This has been proved for bounded convex plane domains. On the other hand, examples show that the conjecture is not true in general. The paper is dedicated to the memory of Vladimir Arnold.