Continuous dependence of the n-th eigenvalue of self-adjoint discrete Sturm-Liouville problems on the problem

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• 作者：Hao Zhu ^{haozhu@mail.sdu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Yuming Shi ; ^{ymshi@sdu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：39A12 ; 34B24 ; 39A70
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 April 2016
• 年：2016
• 卷：260
• 期：7
• 页码：5987-6016
• 全文大小：493 K

文摘

This paper is concerned with continuous dependence of the n-th eigenvalue of self-adjoint discrete Sturm–Liouville problems on the problem. The n-th eigenvalue is considered as a function on the space of the problems, called the n-th eigenvalue function. A necessary and sufficient condition for all the n-th eigenvalue functions to be continuous and several properties of the n-th eigenvalue function on a subset of the space of the problems are given. They play an important role in the study of continuous dependence of the n-th eigenvalue function on the problem. Continuous dependence of the n-th eigenvalue function on the equation and on the boundary condition is studied separately. Consequently, the continuity and discontinuity of the n-th eigenvalue function are completely characterized on the whole space of the problems. Especially, asymptotic behavior of the n-th eigenvalue function near each discontinuity point is given.