Bernard Helffer (University of Paris-Sud)
Abstract: We revisit Courant's nodal domain property for linear combinations of eigenfunctions. This property was proven by Sturm (1836) in the case of dimension 1. Although stated as true for the Dirichlet Laplacian in dimension $ >1$ in a footnote of the celebrated book of Courant-Hilbert (and wrongly attributed to H. Herrmann, a PHD student of R. Courant), it appears to be wrong as first observed by V. Arnold in the seventies. Many counterexamples have been given in collaboration with P. B\'erard in the last years.
To demonstrate in some sense that there is no hope to get such a statement in full generality, and actually any weaker version of the statement,
we prove that there exist metrics $g$ on $\mathbb{T}^2$ (resp. on $\mathbb{S}^2$) which are arbitrarily close to the flat metric (resp. round metric), and an eigenfunction $f$ of the associated Laplace-Beltrami operator such that the set $\{ f \not = 1\}$ has infinitely many connected components. In particular the Extended Courant property is false for these closed surfaces. These results are strongly motivated by a recent paper by Buhovsky, Logunov and Sodin (2019).
As for the positive direction, we prove that the Extended Courant property is true for the isotropic quantum harmonic oscillator in $\mathbb{R}^2$.
This is a work in collaboration with P. B\'erard and P. Charron.
