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

TL;DR

This paper investigates the validity of the Extended Courant property for linear combinations of eigenfunctions on specific domains, providing new examples and insights into when the property holds or fails.

Contribution

The paper introduces two new geometric examples, the equilateral rhombus and the regular hexagon, to analyze the Extended Courant property, expanding understanding beyond previous counterexamples.

Findings

Counterexamples on convex domains confirmed

Extended Courant property fails in new geometric shapes

Numerical computations support theoretical analysis

Abstract

Generalizing Courant's nodal domain theorem, the "Extended Courant property" is the statement that a linear combination of the first $n$ eigenfunctions has at most $n$ nodal domains. In a previous paper (Documenta Mathematica, 2018, Vol. 23, pp. 1561--1585), we gave simple counterexamples to this property, including convex domains. In the present paper, using some input from numerical computations, we pursue the investigation of the Extended Courant property with two new examples, the equilateral rhombus and the regular hexagon.