Eigenvalues of regular Sturm-Liouville problems
type: Item
instance of: scholarly article
title: Eigenvalues of regular Sturm-Liouville problems
zbMATH Open document ID: 0862.34020
author: Anton Zettl
author: Qingkai Kong
published in: Journal of Differential Equations
publication date: 1997-05-26T00:00:00Z
review text: The Sturm-Liouville differential equation \[ -(py')' + qy= \lambda wy \text{ on } (a',b'),\;-\infty \leq a' \leq b' \leq\infty, \;\lambda \in\mathbb{R} \tag{1} \] is considered, where \(p,q,w: (a',b') \to \mathbb{R}\), \(1/p,q,w \in L^1_{\text{loc}} (a',b')\), \(w>0\) a.e. on \((a',b')\). For \(I=[a,b]\), \(a'<a <b <b'\), separated or coupled selfadjoint two-point boundary conditions are taken. For example, the separated boundary conditions can be written in the form \[ \cos \alpha y(a) - \sin \alpha (py')(a) = 0,\;\cos \beta y(b) - \sin \beta (py') (b)=0, \tag{2} \] with parameters \(0\leq \alpha < \pi\), \(0< \beta \leq\pi\). The eigenvalues \(\lambda\) and (normalized) eigenfunctions \(u\) of this boundary eigenvalue problem depend on \(a,b, 1/p,q,w\), and the parameters in the boundary conditions. Therefore, considering \(\lambda\) and \(u\) as functions of these parameters, the question of continuous and differentiable dependence arises. The continuity is shown in Theorem 3.1. This result is somehow mathematical folklore, but no explicit proofs have been known to the authors or the reviewer. Although some results about differentiable dependence can be found in earlier publications, comprehensive results are given for the first time in this paper. Differentiability is proved and explicit formulas for the derivatives with respect to each of the parameters are given. Let us state some of these formulas from Theorems 4.1 and 4.2. We shall assume that \(\lambda\) is a simple eigenvalue and \(u\) a normalized eigenfunction of \(\lambda\) for the boundary eigenvalue problem. Then the derivatives with respect to the indicated parameters are \[ \lambda' (a) = {1\over p(a)} |pu' |^2 (a) - |u|^2 (a) \bigl[q(a) - \lambda (a)w(a) \bigr] \text{ a.e. in } (a',b), \] \[ d\lambda_{(1/p)} (h)= -\int^b_a |pu' |^2h,\;h\in L^1 (a, b). \] If the boundary conditions are separable and written in the form (2), then \(\lambda' (\alpha) = - |u |^2 (a)- |pu' |^2(a)-|pu'|^2(a)\).
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34L99
zbMATH DE Number: 946922
zbMATH Keywords: differentiable dependence of eigenvalues
zbMATH Keywords: Sturm-Liouville problem
zbMATH Keywords: separated boundary conditions
zbMATH Keywords: coupled self-adjoint boundary conditions
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0154
OpenAlex ID: W2037324562
DOI: 10.1006/JDEQ.1996.0154
description: scientific article; zbMATH DE number 946922