Bernard Helffer - Microlocal and Global Analysi...
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Bernard Helffer - Microlocal and Global Analysis, Interactions with Geometry
On Courant's nodal domain property for linear combinations of eigenfunctions (after P. Bérard
and B. Helffer).
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 in dimension > 1. This was rst observed by V. Arnold in the
In this talk, we present simple and explicit counterexamples to this so-called "Herrmann's statement" for domains
in Rd, S2, or T2. We also discuss the existence of a counterexample in a C1, convex domain
in R2 in relation
with the analysis of the number of domains delimited by the level sets of a second eigenfunction for the Neumann
problem. We nally discuss the question to have positive statements. This work has been done in