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 $\mathbb{T}^2$.Comment: 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 accordingl

Bérard, Pierre

Helffer, Bernard

English

arXiv

International audienceAccording 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 $\mathbb{T}^2$

Hal - Université Grenoble Alpes

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

Portail HAL Nantes Université

Hal-Diderot

This version differs notably from arXiv 1803.00449v2 (15 Jun 2018). The part concerning Gelfand's approach to Sturm's 1D theorem has been removed (it is developed in arXiv:1807.03990 and will appear in the Moscow Math. Journal). The treatment of Sturm's 1D theorem in the periodic case has been removed. The short paragraph on product-like constructions has been removed as well. A section concerning the hexagon has been added. Minor changes with respect to hal-01718768, version 3 (7 Jun 2019). The paper focuses on the Extended Courant property for the equilateral rhombus and the regular hexagon. Accepted for publication in a forthcoming  "Erick Balslev Memorial Volume" edited by Sergio Albeverio and Ricardo Weder.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

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

Abstract

Similar works

Full text

Available Versions

Hal - Université Grenoble Alpes

Portail HAL Nantes Université

Hal-Diderot

Archive Ouverte en Sciences de l'Information et de la Communication