Essential spectra of one-dimensional Dirac operators
type: Item
instance of: scholarly article
title: Essential spectra of one-dimensional Dirac operators
zbMATH Open document ID: 1269.34093
author: Shaozhu Chen
author: Jiangang Qi
published in: Integral Equations and Operator Theory
publication date: 2013-03-28T00:00:00Z
review text: The authors introduce a method for calculating the essential spectrum of one-dimensional Dirac operators associated to differential expressions of the form \[ D = W^{-1}\Big\{J\frac{d}{dt} - P \Big\},\quad W=\mathrm{diag}(w_1,w_2), \] where \(w_j(t)>0\) and \(P\) is a self-adjoint \(2\times 2\) matrix of functions. The entries of \(P\) and the \(w_j\) are assumed to be locally integrable on the interval \((a,b)\). In particular, these assumptions permit singularities at the endpoints \(a\) and \(b\). To study the essential spectrum of the operator \(L\) generated by \(D\), the authors introduce two pencils of Sturm-Liouville operators \(\tau_1(\lambda)\) and \(\tau_2(\lambda)\) and determine subsets and supersets for \(\sigma_{\mathrm{ess}}(L)\) using the pencils. For example, setting \[ \sigma_j = \big\{\lambda\in \mathbb R \mid 0\notin \sigma_{\mathrm{ess}}(\tau_j(\lambda))\big\},\quad j=1,2, \] the authors prove \[ \sigma_1\cup \sigma_2 \subset \sigma_{\mathrm{ess}}(L). \] The paper concludes with a section devoted entirely to examples. Each example illustrates how the abstract results developed in the paper can be applied to determine the essential spectrum of various explicit Dirac operators precisely.
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 34B20
zbMATH DE Number: 6149033
zbMATH Keywords: essential spectrum
zbMATH Keywords: Diract operator
zbMATH Keywords: Sturm-Liouville operator
zbMATH Keywords: operator pencil
reviewed by: Roger Nichols
MaRDI profile type: Publication
full work available at URL: s00020-012-1988-2
OpenAlex ID: W2065339690
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DOI: 10.1007/S00020-012-1988-2
description: scientific article; zbMATH DE number 6149033