Eigenvalue problems for Sturm-Liouville equations with transmission conditions
type: Item
instance of: scholarly article
title: Eigenvalue problems for Sturm-Liouville equations with transmission conditions
zbMATH Open document ID: 1073.34096
DOI: 10.1007/BF02916718
author: Xianqiang Yang
published in: Israel Journal of Mathematics
publication date: 2005-03-21T00:00:00Z
review text: For the Sturm-Liouville eigenvalue problem \(-u''+ q(x) u = \lambda u\) for \(x \in (-1, 1), x \not = 0\) subject to the eigenparameter-dependent boundary conditions \(u(-1) = 0, (\lambda - a) u'(1) + \lambda b u(1) = 0\) and the transmission conditions at \(x = 0\): \(u(-0) = h_1 u (+0), u' (-0) = h_2 u' (+0)\), where \(ab > 0, h_1 h_2 > 0\), it was established by the authors [Appl. Math. Comput. 157, 347--355 (2004; Zbl 1060.34007)], that all eigenvalues are real and the eigenfunctions corresponding to different eigenvalues are orthogonal in a Hilbert space. In this article, the asymptotic behavior of the eigenvalues and the corresponding eigenfunctions are given through a realization of the basic solutions which are solutions of the problem on \([-1, 0) \cup (0, 1]\) except that only one boundary condition is satisfied. A counterexample is given to show that \(ab > 0\) and \(h_1 h_2 > 0\) are necessary.
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 34B24
zbMATH DE Number: 2147134
zbMATH Keywords: Sturm-Liouville eigenvalue problem
zbMATH Keywords: eigenparameter-depent boundary conditions
zbMATH Keywords: transmission conditions
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02916718
OpenAlex ID: W1980843314
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description: scientific article; zbMATH DE number 2147134