On Courant's Nodal Domain Property for Linear Combinations of Eigenfunctions. I

Abstract

According to Courant's theorem, an eigenfunction associated 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 Extended Courant Property. In this paper, we propose new, 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 . We also give numerical evidence that the extended Courant property is false for the equilateral triangle with rounded corners, and for the regular hexagon.