Spectral concentration and perturbed discrete spectra
type: Item
instance of: scholarly article
title: Spectral concentration and perturbed discrete spectra
zbMATH Open document ID: 0893.34020
DOI: 10.1016/S0377-0427(97)00173-8
author: J. Ning
published in: Journal of Computational and Applied Mathematics
publication date: 1998-02-09T00:00:00Z
review text: The authors consider the Sturm-Liouville equation \[ -y''+ q(x)y= \lambda y\;(0\leq x<\infty), \quad u(0)=0, \] where the potential is either \[ q(x)= -cx^2+x \] or \[ q(x)= -cx-(1+x)^{-1} \] with a parameter \(c\geq 0\). It is known that the spectrum for \(c>0\) is continuous, whereas for \(c=0\) it is discrete in \((-\infty, \infty)\) and \((-\infty,0)\), respectively. The influence of the eigenvalues \(\lambda_n\) for \(c=0\) on the spectral function \(\rho_c\) for small \(c>0\) is investigated. It turns out that, as \(c\downarrow 0\), ``spectral concentration'' occurs successively near \(\lambda_0, \lambda_1, \lambda_2 \dots\), which is reflected by local maxima of the density \(\rho_c'\). The main technical ingredient is a new representation formula for \(\rho_c'\) in terms of a solution to a first order ODE.
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34L05
zbMATH DE Number: 1115946
zbMATH Keywords: perturbed discrete spectra
zbMATH Keywords: Titchmarsh-Weyl function
zbMATH Keywords: spectral function
zbMATH Keywords: spectral concentration
zbMATH Keywords: Sturm-Liouville equation
reviewed by: Frank Mantlik
MaRDI profile type: Publication
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full work available at URL: s0377-0427(97)00173-8
OpenAlex ID: W1992436140
description: scientific article; zbMATH DE number 1115946