On eigenvalues of the one-dimensional Dirac operator with oscillatory decreasing potential
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)].
scientific article; zbMATH DE number 6115297
scientific article