On the characterization of the smoothness of skew-adjoint potentials in periodic Dirac operators
type: Item
instance of: scholarly article
title: On the characterization of the smoothness of skew-adjoint potentials in periodic Dirac operators
zbMATH Open document ID: 1191.47060
author: T. Kappeler
author: Frédéric Serier
author: Peter Topalov
published in: Journal of Functional Analysis
publication date: 2009-05-12T00:00:00Z
review text: Let \[ L(\varphi)u=i\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix} u'+ \begin{pmatrix} 0 & \varphi_1 \\ \varphi_2 & 0 \end{pmatrix} u \] be the linear operator acting on \(u\in H^1_{\text{loc}}([0,2],{\mathbb C}^2)\) subject to periodic boundary conditions, where the potential \(\varphi=(\varphi_1,\varphi_2)\) is skew symmetric in the sense that \(\varphi_2=-\overline{\varphi_1}\). In the literature, \(L(\varphi)\) is often referred to either as a `Dirac' or a `Zakharov-Shabat' operator. The main objective of this paper is to establish precise connections between asymptotics of the periodic spectrum of \(L(\varphi)\) and regularity properties (in suitable weighted Sobolev spaces) of \(\varphi\).
reviewed by: Lyonell Boulton
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 5553024
zbMATH Keywords: focusing NLS
zbMATH Keywords: periodic Dirac operators
zbMATH Keywords: decay of gap lengths
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.01.027
OpenAlex ID: W2028237508
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DOI: 10.1016/J.JFA.2009.01.027
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description: scientific article; zbMATH DE number 5553024