Courant's nodal domain theorem for positivity preserving forms
type: Item
instance of: scholarly article
title: Courant's nodal domain theorem for positivity preserving forms
zbMATH Open document ID: 1444.47004
DOI: 10.4171/JST/292
author: Matthias Keller
author: Michael Schwarz
published in: Journal of Spectral Theory
publication date: 2020-05-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1712.07414
review text: In [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1923, 81--84 (1923; JFM 49.0342.01)] \textit{R. Courant} proved a result about the nodal domains of the eigenfunctions of a self-adjoint differential operator associated to a quadratic form on a bounded domain of \(\mathbb{R}^d\) with Dirichlet boundary conditions. A nodal domain is a maximal connected set where an eigenfunction does not change its sign. The authors first introduce a notion of nodal domains for positivity preserving forms with compact resolvent, i.e., quadratic forms on an \(L^2\)-space associated to self-adjoint compact resolvents \((L+I)^{-1}\) that map positive functions to positive functions. This notion covers Courant's one and other variations in the literature. Then the authors establish a Courant-type result for these forms.
reviewed by: Rosario Corso
Mathematics Subject Classification ID: 47A07
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 7200137
zbMATH Keywords: nodal domain
zbMATH Keywords: eigenfunction
zbMATH Keywords: Dirichlet form
zbMATH Keywords: compact resolvent
MaRDI profile type: Publication
arXiv ID: 1712.07414
description: scientific article