Quadratic growth of convergence radii for eigenvalues of two-parameter Sturm-Liouville equations
type: Item
instance of: scholarly article
title: Quadratic growth of convergence radii for eigenvalues of two-parameter Sturm-Liouville equations
zbMATH Open document ID: 0853.34031
published in: Journal of Differential Equations
publication date: 1996-07-02T00:00:00Z
review text: The paper deals with two-parameter Sturm-Liouville problem \(y'' + (\mu + \lambda r) y = 0\) on a finite interval \([a,b]\) with selfadjoint boundary conditions \(\alpha_1 y(a) + \alpha_2 y'(a) = 0\), \(\beta_1 y(b) + \beta_2 y'(b) = 0\). Here \(\lambda\) and \(\mu\) are parameters and the function \(r\) is continuous and real-valued. The \(n\)-th eigenvalue \(\mu_n (\lambda)\) of such a problem can be expanded in powers of \(\lambda\) for sufficiently small \(|\lambda |\). It is proved that the convergence radii \(\rho_n\) corresponding to these expansions are of order \(O(n^2)\) if \(r\) is analytic on \([a,b]\). Here the Liouville transformation is applied to the given problem. In addition, concrete lower bounds for \(\liminf_{ n \to \infty} (\rho_n/n^2)\) are given. The results apply nicely to the Mathieu equation and the two-parameter Airy equation.
reviewed by: Christiane Tretter
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34E05
zbMATH DE Number: 895814
zbMATH Keywords: two-parameter Sturm-Liouville problem
zbMATH Keywords: eigenvalue
zbMATH Keywords: convergence radii
zbMATH Keywords: expansions
zbMATH Keywords: Liouville transformation
zbMATH Keywords: Mathieu equation
zbMATH Keywords: two-parameter Airy equation
author: Hans W. Volkmer
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0098
OpenAlex ID: W1975360288
DOI: 10.1006/JDEQ.1996.0098
description: scientific article; zbMATH DE number 895814