Eigenvalue ratios for Sturm--Liouville operators
type: Item
instance of: scholarly article
title: Eigenvalue ratios for Sturm--Liouville operators
zbMATH Open document ID: 0785.34027
author: Mark S. Ashbaugh
author: Rafael D. Benguria
published in: Journal of Differential Equations
publication date: 1993-10-18T00:00:00Z
review text: This paper deals with estimates for eigenvalue ratios for regular Sturm-Liouville problems \(-[p(x)y']'+q(x)y=\lambda w(x)y\) on a finite interval with Dirichlet boundary conditions. In the general case \(q \geq 0\), the authors prove the upper estimate \(\lambda_ m/ \lambda_ l \leq K \{m/l\} ^ 2/k\) for \(m>l \geq 1\) where \(k\), \(K \geq 0\) are such that \(k \leq pw \leq K\) and \(\{ m/l \}\) denotes the least integer greater or equal \(m/l\). This result generalizes estimates for the Dirichlet eigenvalues of the one-dimensional Schrödinger equation \(-y''+q(x)y=\lambda y\). For \(q=0\), it can be improved and supplemented by a lower estimate to \(km^ 2/Kl^ 2 \leq \lambda_ m/ \lambda_ l \leq Km^ 2/kl^ 2\). In addition the upper bound for the eigenvalue ratio of the first two eigenvalues is improved using a different approach based on the authors' proof of the Payne-Pólya-Weinberger conjecture. Finally, the equation of motion of the inhomogeneous stretched string \(-y''=\lambda w(x)y\) is used to compare the best bounds obtained for \(\lambda_ 2/ \lambda_ 1\) to numerically computed results of Keller and of Mahar and Willner.
reviewed by: Christiane Tretter
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34L40
zbMATH DE Number: 425453
zbMATH Keywords: estimates for eigenvalue ratios
zbMATH Keywords: regular Sturm-Liouville problems
zbMATH Keywords: one-dimensional Schrödinger equation
zbMATH Keywords: Payne-Pólya-Weinberger conjecture
Wikidata QID: Q59158288
MaRDI profile type: Publication
full work available at URL: jdeq.1993.1047
OpenAlex ID: W2090945912
DOI: 10.1006/JDEQ.1993.1047
description: scientific article; zbMATH DE number 425453