Singular left-definite Sturm--Liouville problems
type: Item
instance of: scholarly article
title: Singular left-definite Sturm--Liouville problems
zbMATH Open document ID: 1058.34029
author: Hongyou Wu
author: Anton Zettl
author: Qingkai Kong
published in: Journal of Differential Equations
publication date: 2004-11-18T00:00:00Z
review text: This paper deals with the eigenvalues of singular left-definite Sturm-Liouville problems for the equation \[ My:=- (py')'+qy= \lambda wy\quad\text{on}\quad J=(a,b),\qquad -\infty\leq a<b \leq\infty,\tag{1} \] where \[ (i)\quad 1/p,\,q,\,w\in L^{1}_{\text{ loc}}(J;\mathbb{R}), \qquad(ii)\quad p,\,|w|>0\text{ a.e. on }J,\qquad (iii)\quad w\text{ changes sign on }J. \] Since the weight function \(w\) changes sign, the standard Sturm-Liouville theory cannot be applied to this equation. The authors eliminate this difficulty by the following observation. For \(\lambda\in \mathbb{R}\), let \(S(\lambda)\) be a selfadjoint realization of the differential expression \[ \frac{1}{|w|} (M-\lambda w) \] in the Hilbert space \(H=L^{2}(J,|w|)\) equipped with the inner product \((f,g)=\int_Jf \bar{g}|w|\). By \(D\) we denote the domain of \(S(\lambda)\). Then \(\lambda\) is in the spectrum of \((1)\) on \(D\) if and only if \(0\in\sigma (S(\lambda))\). Among other results the authors obtain the following assertion. Assume that the problem \((1)\) on \((D)\) is left-definite, i.e.,there exists \(c>0\) such that \(\int_J(Mf)\bar{f}\geq c\) for all \(f\) in \(D\cap U\), where \(U\) is the unit ball of \(H\). For \(n\in\mathbb{N}_{0}=\{0,1,2,\ldots\}\), we put \[ \xi_{n}(\lambda)=\inf_{F\in G_{n+1}(D)}\sup_{f\in F\cap U}(S(\lambda)f,f), \] where \(G_{n+1}(D)\) is the set of \(n\)-dimensional subspaces of \(D\). For each \(n\in\mathbb{N}_{0}\), the equation \(\xi_{n}(\lambda)=0\) always has exactly one positive root \(\lambda=\lambda_{n}\) and one negative root \(\lambda=\lambda_{-n}\), and they satisfy that \(|\lambda_{\pm n}|\geq\xi_{n}(0)\). (i) Suppose that for some \(n\in\mathbb{N}_{0}\), \(0\) is an eigenvalue of \(S(\lambda_{n})\). Then \(\lambda_{j}\), \(j=0,\ldots,n\), are eigenvalues of problem \((1)\) on \(D\) and satisfy \[ 0<\lambda_{0}\leq\lambda_{1}\leq\cdots\leq\lambda_{n}.\tag{2} \] Moreover, these eigenvalues are the only points of the spectrum of \((1)\) on \(D\) in \(…
reviewed by: Kazushi Yoshitomi
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 34L15
zbMATH DE Number: 2116216
zbMATH Keywords: singular problems
zbMATH Keywords: left-definite
zbMATH Keywords: eigenvalues
zbMATH Keywords: eigenfunctions
zbMATH Keywords: eigenvalue inequalities
MaRDI profile type: Publication
full work available at URL: j.jde.2004.07.027
OpenAlex ID: W1995040889
cites work: Algorithm 810: The SLEIGN2 Sturm-Liouville Code
cites work: Q3333353
cites work: Asymptotics of eigencurves for second order ordinary differential equations. I
cites work: Corrigendum: Spectral properties of two parameter eigenvalue problems II
cites work: Left definite Sturm-Liouville problems with eigenparameter dependent boundary conditions
cites work: Eigencurves for Two-Parameter Sturm-Liouville Equations
cites work: On the spectrum of second-order differential operators with complex coefficients
cites work: A Krein space approach to symmetric ordinary differential operators with an indefinite weigth function
cites work: Sturm-Liouville operators with an indefinite weight function
cites work: Q5336643
cites work: Inequalities among eigenvalues of Sturm-Liouville problems
cites work: Q4056365
cites work: Some remarks on the strong limit-point condition of second-order linear differential expressions
cites work: Singular Sturm-Liouville problems with nonnegative and indefinite weights
cites work: Indefinite Sturm–Liouville problems
cites work: Q4940921
cites work: Left-definite Sturm-Liouville problems
cites work: Discreteness conditions for the spectrum of ordinary differential operators
cites work: A general left-definite theory for certain self-adjoint operators with applications to differential equations
cites work: Q4776206
cites work: Linksdefinite singuläre kanonische Eigenwertprobleme. II.
cites work: Singular Sturm-Liouville Problems: The Friedrichs Extension and Comparison of Eigenvalues
cites work: Complementary remarks about the limit point and limit circle theory
cites work: Q3242330
cites work: A left-definite study of Legendre's differential equation and the fourth-order Legendre type differential equation
cites work: Q4876631
cites work: The virial theorem and its application to the spectral theory of Schrödinger operators
cites work: Q4350113
DOI: 10.1016/J.JDE.2004.07.027
Recommended article: Left-definite spaces of singular Sturm-Liouville problems
Recommended article: Left-definite Sturm-Liouville problems
Recommended article: On singular Sturm–Liouville boundary-value problems
Recommended article: Strongly singular Sturm-Liouville problems
Recommended article: Singular left-definite boundary value problems
Recommended article: On a class of singular Sturm–Liouville problems
Recommended article: Singular self-adjoint Sturm-Liouville problems
Recommended article: The singular multiparameter Sturm-Liouville problem
Recommended article: Singular Sturm-Liouville problems with nonnegative and indefinite weights
Recommended article: Singular sturm liouville problems and existence of solutions to singular nonlinear boundary value problems
description: scientific article; zbMATH DE number 2116216