Spectral function of Krein's and Kotani's string in the class \(\Gamma\)
type: Item
instance of: scholarly article
title: Spectral function of Krein's and Kotani's string in the class \(\Gamma\)
zbMATH Open document ID: 1284.47031
author: Yuji Kasahara
published in: Proceedings of the Japan Academy. Series A
publication date: 2013-04-25T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1354802416
review text: The author focuses on the asymptotic behavior of the spectral function associated with the generalized second-order Sturm-Liouville operator \[ \mathcal{L}=\frac{\mathrm{d}}{\mathrm{d}m(x)}\frac{\mathrm{d}}{\mathrm{d}x},\quad -\infty<x<\ell, \] where \(m\) is a string, i.e., a nondecreasing and right-continuous function \(m:(-\infty,\infty)\to[0,+\infty]\) satisfying \(m(-\infty+0)=0\), the Lebesgue-Stieltjes measure \(\mathrm{d}m\) describes the mass-distribution of the string \(m\), and \(\ell=\sup\{x: m(x)<\infty\}\). Note that the classical Sturm-Liouville operator \(a(x)\frac{\mathrm{d}^2}{\mathrm{d}x}+b(x)\frac{\mathrm{d}}{\mathrm{d}x}\) can be written as \(\mathcal{L}\). Necessary and sufficient conditions guaranteeing that the spectral function varies regularly with index~\(1\) are established as main results. These results are closely related to the so-called de~Haan theory. Two illustrative examples are also provided.
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60G51
zbMATH DE Number: 6157749
zbMATH Keywords: generalized Sturm-Liouville operator
zbMATH Keywords: spectral measure
zbMATH Keywords: diffusion
zbMATH Keywords: Krein's correspondence
zbMATH Keywords: de Haan theory
reviewed by: Petr Zemánek
MaRDI profile type: Publication
OpenAlex ID: W1977603300
cites work: Q4726487
cites work: Brownian representation of a class of Lévy processes and its application to occupation times of diffusion processes
cites work: Krein's strings with singular left boundary
cites work: Q3959913
DOI: 10.3792/PJAA.88.173
description: scientific article; zbMATH DE number 6157749