On the missing eigenvalue problem for Dirac operators
type: Item
instance of: scholarly article
title: On the missing eigenvalue problem for Dirac operators
zbMATH Open document ID: 1390.34047
author: Yuping Wang
author: V. A. Yurko
published in: Applied Mathematics Letters
publication date: 2018-05-11T00:00:00Z
review text: The authors study the Dirac system in the form \[ \begin{cases} u_2' + V_1(x) u_1 = \lambda u_1, \\ - u_1' + V_2(x) u_2 = \lambda u_2, \end{cases} \quad x \in (0, \pi), \] with the boundary conditions \[ \begin{aligned} u_1(0, \lambda) \sin \alpha + u_2(0, \lambda) \cos \alpha = 0, \\ u_1(\pi, \lambda) \sin \beta + u_2(\pi, \lambda) \cos \beta = 0, \end{aligned} \] where \(0 \leq \alpha, \beta < \pi\), \(V_1\) and \(V_2\) are absolutely continuous real-valued functions. Two inverse problems are investigated: the Borg-type problem consists in recovering the potentials \(V_1\) and \(V_2\) from two spectra (the second spectrum corresponds to the boundary conditions with \(\beta_0\) instead of \(\beta\), \(\beta \neq \beta_0\)), and the Hochstadt-Lieberman-type problem consists in recovering \(V_1\) and \(V_2\) from one spectrum under the assumption that \(V_1\) and \(V_2\) are known a priori on \([\pi/2, \pi]\). In both cases, the coefficients \(\alpha\) and \(\beta\) are assumed to be known a priori. The authors prove that the potentials are uniquely defined by two spectra with a missing eigenvalue in the Borg case, and by one spectrum with a missing eigenvalue in the Hochstadt-Lieberman case.
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 34L40
zbMATH DE Number: 6867975
zbMATH Keywords: inverse spectral problem
zbMATH Keywords: potential
zbMATH Keywords: Dirac operator
zbMATH Keywords: missing eigenvalue problem
reviewed by: Natalia Bondarenko
MaRDI profile type: Publication
full work available at URL: j.aml.2018.01.004
OpenAlex ID: W2783926603
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DOI: 10.1016/J.AML.2018.01.004
description: scientific article; zbMATH DE number 6867975