Eigenvalues variation. I: Neumann problem for Sturm--Liouville operators
type: Item
instance of: scholarly article
title: Eigenvalues variation. I: Neumann problem for Sturm--Liouville operators
zbMATH Open document ID: 0784.34021
author: Monique Dauge
author: Bernard Helffer
published in: Journal of Differential Equations
publication date: 1993-10-25T00:00:00Z
review text: The authors investigate variations with respect to the domain, of eigenvalues for Neumann problems in one dimensions of the type \(D_ x(p(x)D_ x u(x))+ q(x)u(x)=f(x)\), \(0<x<a\), \(u'(0)=0\), \(u'(a)=0\). They prove formula (F) which relates the potential \(q\) and the first derivative of the eigenvalues \(\lambda\) with respect to the domain. Equation (F) is a differential equation which has the form: (F) \(\lambda'=c(q-\lambda)\) where \(c\) is a positive function. Together with asymptotics when the width of the domain goes to zero or to infinity, they derive from this formula precise information about the variation of the eigenvalues. In a second paper [ibid. 104, 244-278 (1993)], the authors study the same problems for operators which arise in mathematical physics.
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34L05
zbMATH DE Number: 431261
zbMATH Keywords: Neumann problems
zbMATH Keywords: variation of the eigenvalues
zbMATH Keywords: operators
zbMATH Keywords: mathematical physics
reviewed by: Pierre Bolley
MaRDI profile type: Publication
full work available at URL: jdeq.1993.1071
OpenAlex ID: W1645807286
DOI: 10.1006/JDEQ.1993.1071
description: scientific article; zbMATH DE number 431261