Oscillation results for Sturm--Liouville problems with an indefinite weight function
type: Item
instance of: scholarly article
title: Oscillation results for Sturm--Liouville problems with an indefinite weight function
zbMATH Open document ID: 1078.34011
author: P. A. Binding
author: Heinz Langer
author: Manfred Möller
published in: Journal of Computational and Applied Mathematics
publication date: 2004-08-10T00:00:00Z
review text: Let \(q\) and \(r\) be real-valued functions in \(L^1(0,l)\) with \(| r| >0\) a.e.\ and consider the Sturm-Liouville problem \[ -y''(x) + q(x) y(x) = \lambda r(x) y(x) \qquad\text{ on}\qquad (0,l), \qquad y'(0) = y'(l) = 0. \] When \(r>0\) a.e., denote the eigenvalues by \(\lambda_1<\lambda_2<\dots\) and the corresponding eigenfunctions by \(y_n\). Then by the classical Sturm oscillation theory, the number of zeros (or the oscillation count) of \(y_n\) in \((0,l)\) is equal to \(n-1\). When the weight \(r\) is indefinite (i.e., takes both signs on sets of positive measure), the spectrum consists of two sequences of real eigenvalues tending to \(-\infty\) and \(+\infty\), and, possibly, finitely many pairs of complex conjugate nonreal eigenvalues; moreover, a finite number of eigenvalues might be nonsimple. The main aim of the paper is to establish a formula for the oscillating count \(\omega_n\) of the eigenfunction \(y_n\) corresponding to the \(n\)th positive eigenvalue \(\lambda_n\) of the above Sturm-Liouville problem in the indefinite weight case. The oscillation count \(\omega_n\) is shown to depend on the algebraic multiplicities of the eigenvalues and their signatures. The proof exploits the Titchmarsh-Weyl \(m\)-function and the Prüfer angle method.
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34B09
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 47B50
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34L20
zbMATH DE Number: 2085586
zbMATH Keywords: Sturm--Liouville operator
zbMATH Keywords: indefinite weight
zbMATH Keywords: oscillation
reviewed by: R. Hryniv
describes a project that uses: SLEIGN2
MaRDI profile type: Publication
full work available at URL: j.cam.2004.01.015
OpenAlex ID: W2165735049
Wikidata QID: Q124629927
cites work: Q5555880
cites work: On the m -Coefficient of Weyl for a Differential Equation with an Indefinite Weight Function
cites work: Algorithm 810: The SLEIGN2 Sturm-Liouville Code
cites work: Oscillation theory for Sturm-Liouville problems with indefinite coefficients
cites work: A Krein space approach to symmetric ordinary differential operators with an indefinite weigth function
cites work: Q4869257
cites work: On a Property of the M -Coefficient of a Secondorder Linear Differential Equation
cites work: Orthogonal Systems of Eigenvectors and Associated Vectors for Symmetric Holomorphic Operator Functions
cites work: Spectral theory of ordinary differential operators
DOI: 10.1016/J.CAM.2004.01.015
description: scientific article; zbMATH DE number 2085586