Fredholm properties of radially symmetric, second order differential operators
type: Item
instance of: scholarly article
title: Fredholm properties of radially symmetric, second order differential operators
zbMATH Open document ID: 1246.47011
DOI: 10.1504/IJDSDE.2011.041878
author: Alin Pogan
author: Arnd Scheel
published in: International Journal of Dynamical Systems and Differential Equations
publication date: 2011-11-17T00:00:00Z
review text: The authors analyze the Fredholm properties of radially symmetric elliptic operators \[ \begin{aligned} \mathcal{L}_{\text{rad}}&=D(r)\left(\frac{d^2}{dr^2}+\frac{k-1}{r}\frac{d}{dr}\right)+Q(r)\frac{d}{dr}+R(r),\\ \mathcal{L}_{\text{rad}}^l&=D(r)\left(\frac{d^2}{dr^2}+\frac{k-1}{r}\frac{d}{dr}-\frac{l^2}{r^2}\right)+R(r).\end{aligned} \] They show that the Fredholm index of either \(\mathcal{L}_{\text{rad}}\) or \(\mathcal{L}_{\text{rad}}^l\) is related to the Morse index of the operator at infinity under some suitable assumptions and, as a consequence, linear operators are Fredholm in exponentially weighted spaces for almost all weights. The result provides a basic tool for the analysis of perturbation and bifurcation problems in the presence of essential spectrum. A simple example is given to illustrate the application of the main results.
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 5973366
zbMATH Keywords: differential operators
zbMATH Keywords: Fredholm properties
zbMATH Keywords: radial symmetry
zbMATH Keywords: bifurcation
zbMATH Keywords: essential spectrum
zbMATH Keywords: far-field matching
zbMATH Keywords: trimolecular chemical reaction-diffusion systems
reviewed by: Peixuan Weng
MaRDI profile type: Publication
full work available at URL: ijdsde.2011.041878
OpenAlex ID: W2075370535
description: scientific article; zbMATH DE number 5973366