On Courant's nodal domain property for linear combinations of eigenfunctions, Part I

Abstract

According to Courant's theorem, an eigenfunction as\-sociated with the $n$-th eigenvalue $\lambda \_n$ has at most $n$ nodal domains. A footnote in the book of Courant and Hilbert, states that the same assertion is true for any linear combination of eigenfunctions associated with eigenvalues less than or equal to $\lambda \_n$. We call this assertion the \emph{Extended Courant Property}.\smallskipIn this paper, we propose simple and explicit examples for which the extended Courant property is false: convex domains in ${\mathbb{R}}^n$ (hypercube and equilateral triangle), domains with cracks in ${\mathbb{R}}^2$, on the round sphere ${\mathbb{S}}^2$, and on a flat torus ${\mathbb{T}}^2$.

To appear in Documenta Mathematica.Modifications with respect to version 4: Introduction rewritten. To shorten the paper two sections (Section 7, Numerical simulations and Section 8, Conjectures) have been removed and will be published elsewhere. Related to the paper arXiv:1803.00449v2. Small overlap with arXiv:1803.00449v1 which will be modified accordingly

References in corpus (2)

• On the number of connected components of random algebraic hypersurfaces
• Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle

Cited by in corpus (2)

• Sturm's theorem on zeros of linear combinations of eigenfunctions
• Level sets of certain Neumann eigenfunctions under deformation of Lipschitz domains. Application to the Extended Courant Property