Inverse spectral problems for left-definite Sturm-Liouville equations with indefinite weight
type: Item
instance of: scholarly article
title: Inverse spectral problems for left-definite Sturm-Liouville equations with indefinite weight
zbMATH Open document ID: 1019.34030
DOI: 10.1016/S0022-247X(02)00123-3
author: Patrick J. Browne
author: P. A. Binding
author: Bruce A. Watson
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-11-28T00:00:00Z
review text: The authors consider the Sturm-Liouville problem of the form \[ -(py')'+qy=\lambda ry \quad \text{on }[a,b], \tag{1} \] \[ y(a)\cos\alpha+(py')(a)\sin\alpha = 0, \quad \alpha\in \left(-{\pi\over 2}, {\pi\over 2}\right],\tag{2} \] \[ y(b)\cos\beta+(py')(b)\sin\beta = 0, \quad \beta\in \left(-{\pi\over 2}, {\pi\over 2}\right], \tag{3} \] assuming that \(p\) is a \(C^1\) positive function with absolutely continuous derivative on \([a,b]\) with \(a<0<b\); \(r\) is a \(C^1\) positive function with absolutely continuous derivative on \((0,b]\) and a negative function with absolutely continuous derivative on \([a,0)\). It is assumed also that \(-\infty<r(0^-)<0<r(0^+)<\infty\) or that \(r\) is differentiable at \(0\) with \(r'(0)>0\), and \(q\) and \(\widetilde{q}\) are real \(L^1\) functions on \([a,b]\). The main result is as follows: Let \(\lambda_n\) and \(\omega_n\) denote the eigenvalues and the eigenfunctions of problem (1)-(3), where \(\omega_n\) is normalized by \(\omega_n(a)=\sin\alpha\), \(\omega'_n(a)=-\cos\alpha\). Let \(\rho_n=|||\omega_n|||=\sqrt{\int_a^b \overline{\omega_n}(-(p\omega'_n)'+q\omega_n)dx}\). Define \(\widetilde{\rho}_n\) and \(\widetilde{\lambda}_n\) in an analogous manner but with \(q\) replaced by \(\widetilde{q}\). Suppose that \(\lambda_n=\widetilde{\lambda}_n\) and \(\rho_n=\widetilde{\rho}_n\) for all \(n=\pm 1, \pm 2,\dots\). Then \(q=\widetilde{q}\).
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 1836761
zbMATH Keywords: indefinite weight
zbMATH Keywords: left-definite Sturm-Liouville equations
zbMATH Keywords: inverse spectral equations
reviewed by: V. Pivovarchik
MaRDI profile type: Publication
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full work available at URL: s0022-247x(02)00123-3
OpenAlex ID: W2087313088
description: scientific article; zbMATH DE number 1836761