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

TL;DR

This paper investigates the validity of the Extended Courant Property, which suggests linear combinations of eigenfunctions should have at most as many nodal domains as the eigenfunction with the corresponding eigenvalue, providing explicit counterexamples.

Contribution

The paper presents explicit counterexamples in various domains demonstrating the failure of the Extended Courant Property, challenging a long-standing assumption in spectral theory.

Findings

Counterexamples in convex domains, spheres, and tori showing failure of the property

Explicit constructions of linear combinations with more nodal domains than expected

Implications for spectral theory and eigenfunction analysis

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 $\R^n$ (hypercube and equilateral triangle), domains with cracks in $\mathbb{R}^2$, on the round sphere $\mathbb{S}^2$, and on a flat torus $\mathbb{T}^2$.