The asymptotic nature of spectral functions in Sturm-Liouville problems with continuous spectrum
type: Item
instance of: scholarly article
title: The asymptotic nature of spectral functions in Sturm-Liouville problems with continuous spectrum
zbMATH Open document ID: 0895.34021
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-09-21T00:00:00Z
review text: The author considers the spectral function \(\rho_\alpha (\mu)\) associated with the Sturm-Liouville equation \[ y''(x)+ \bigl[\lambda- q(x)\bigr] y(x)=0 \] and the boundary condition \(y(0)\cos \alpha+ y'(0)\sin \alpha=0\). The asymptotic behavior of \(\rho_\alpha (\mu)\) as \(\mu\to \infty\) is discussed and an explicit formula is given for the Dirichlet case, \(\alpha=0\), which has not been covered previously. Similar formulas can be obtained with the same methods for all values of \(\alpha\) and hence this paper generalizes results obtained by Atkinson, Harris, Levitan, and Marçenko.
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34B20
zbMATH DE Number: 1115566
zbMATH Keywords: spectral functions
zbMATH Keywords: Sturm-Liouville problems
author: M. S. P. Eastham
reviewed by: Qingkai Kong
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5560
OpenAlex ID: W2053799428
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DOI: 10.1006/JMAA.1997.5560
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description: scientific article; zbMATH DE number 1115566