On eigenvalues of the one-dimensional Dirac operator with oscillatory decreasing potential
type: Item
instance of: scholarly article
title: On eigenvalues of the one-dimensional Dirac operator with oscillatory decreasing potential
zbMATH Open document ID: 1263.34124
author: Pavel Nesterov
published in: Mathematical Physics, Analysis and Geometry
publication date: 2012-12-12T00:00:00Z
review text: The author proves results concerning localization of the set of eigenvalues for the self-adjoint one-dimensional Dirac operator \[ D = \left( \begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix} \right) \frac{d}{d x} + U (x), \] considered acting in \(L_{2} ((0, \infty); \mathbb{C}^{2}),\) in the case when the entries of the potential \(U( x)\) are real-valued functions of the form \(P (x)/(x + 1)^{\alpha}\) with \(0 < \alpha \leq 1\), \(P (x)\) being either a periodic function or a trigonometric polynomial \[ P (x) = \sum_{j = - N}^{N} p_{j} e^{i \omega_{j} x}, \] \(p_{j} \in \mathbb{C}\), \(\omega_{j} \in \mathbb{R}\), \(\rho_{j} = \bar{p}_{- j}\) and \(\omega_{- j} = - \omega_{j} \;(j = 0, \pm 1, \dotsc, \pm N)\). The approach is based on the technique of asymptotic integration of linear systems of differential equations proposed by \textit{W. A. Harris jun.} and \textit{D. A. Lutz} [J. Math. Anal. Appl. 48, 1--16 (1974; Zbl 0304.34043); J. Math. Anal. Appl. 51, 76--93 (1975; Zbl 0315.34070)].
reviewed by: Petru A. Cojuhari
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 34B09
zbMATH DE Number: 6115297
zbMATH Keywords: Dirac operator
zbMATH Keywords: oscillatory decreasing potential
zbMATH Keywords: asymptotic integration
zbMATH Keywords: method of averaging
zbMATH Keywords: Levinson's theorem
MaRDI profile type: Publication
full work available at URL: s11040-012-9114-x
OpenAlex ID: W1971842261
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DOI: 10.1007/S11040-012-9114-X
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description: scientific article; zbMATH DE number 6115297