Matrix finite-zone Dirac-type equations
type: Item
instance of: scholarly article
title: Matrix finite-zone Dirac-type equations
zbMATH Open document ID: 1019.34026
published in: Journal of Functional Analysis
publication date: 2002-11-28T00:00:00Z
review text: Consider the problem \[ \frac{dw}{dx}=(izj + \xi(x))w \tag{1} \] on the half-line \((0,\infty)\) and the line \((-\infty ,\infty)\). Here, \(\xi(x)\), \(w(x,z)\) and \(j\) are \(n\times n\)-matrices (with \(n=2m\)) of the form \[ j=\begin{pmatrix} I_m&0\\ 0&-I_m\end{pmatrix},\qquad \xi(x)=\begin{pmatrix} 0&q(x)\\q^*(x)&0\end{pmatrix}, \] where \(q\) is assumed to be locally bounded and \(z\) is the spectral parameter. In the case of the half-line, the boundary condition \(w(0,z)=I_n\) is imposed. In that case, \(\varphi (z)\) is called a Weyl-Titchmarsh matrix function for (1) if \[ \int_0^\infty \begin{pmatrix} \varphi ^*(z)& I_m\end{pmatrix} w^*(x,z)w(x,z) \begin{pmatrix} \varphi (z)\\ I_m\end{pmatrix} dx<\infty . \] It is known that there is a unique Weyl-Titchmarsh matrix function. The case \((-\infty ,\infty)\) is reduced to the half-line case. Assuming that the Weyl-Titchmarsh matrix function has the form \[ R(z)^{-\frac 12}\begin{pmatrix} iP(z)& Q_1(z)\\ Q_2(z)&iS(z)\end{pmatrix} \tag{2} \] with matrix polynomials \(P\), \(Q_1\), \(Q_2\), \(S\), and a scalar polynomial \(R\), it is shown that they have to satisfy certain conditions. Finally, under condition (2) and in the case \(m=1\), a procedure to recover the potential \(q\) from the Weyl-Titchmarsh function is given. In particular, \(q\) is quasiperiodic.
Mathematics Subject Classification ID: 34B20
Mathematics Subject Classification ID: 34A55
zbMATH DE Number: 1836683
zbMATH Keywords: Titchmarsh-Weyl matrix function
zbMATH Keywords: Dirac equation
zbMATH Keywords: quasiperiodic potential
zbMATH Keywords: inverse problem
author: Lev A. Sakhnovich
MaRDI profile type: Publication
full work available at URL: jfan.2002.3956
OpenAlex ID: W2088602743
cites work: Q3925229
cites work: Q5556776
cites work: Matrix finite-zone operators
cites work: Q3997293
cites work: Q3834554
cites work: The spectrum of Hill's equation
cites work: Q3760860
cites work: Spectral theory of canonical differential systems. Method of operator identities
DOI: 10.1006/JFAN.2002.3956
description: scientific article; zbMATH DE number 1836683