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On the classes of explicit solutions of Dirac, dynamical Dirac and Dirac–Weyl systems with non-vanishing at infinity potentials, their properties and applications
Journal of Differential Equations ( IF 2.4 ) Pub Date : 2021-02-01 , DOI: 10.1016/j.jde.2020.11.037
Alexander Sakhnovich

Abstract We construct explicitly potentials, Darboux matrix functions and corresponding solutions of Dirac, dynamical Dirac and Dirac–Weyl systems using generalised Backlund-Darboux transformation (GBDT) in the important case of nontrivial initial systems. In this way, we construct explicit solutions of systems with non-vanishing at infinity potentials, including steplike and power-law growth potentials. Thus, the constructed potentials (systems) differ fundamentally from the actively studied case of GBDT for the trivial initial systems. Generalised matrix eigenvalues A (not necessarily diagonal) and corresponding generalised matrix eigenfunctions Π ( x ) of the nontrivial initial systems are used in the GBDT constructions in this paper. Explicit expressions for these Π ( x ) are new and the method of deriving these expressions may be applied to various other important problems. The case of Dirac–Weyl system, which is of interest in electron dynamics and graphene theory is studied in greater detail, and generalised separation of variables appears in our approach to the study of this system. Explicit expressions for Weyl–Titchmarsh functions (in the form of pseudo-realisations) are derived.

中文翻译:

关于在无穷势能上不为零的狄拉克、动力学狄拉克和狄拉克-外尔系统的显式解的类、它们的性质和应用

摘要 我们在非平凡初始系统的重要情况下使用广义 Backlund-Darboux 变换 (GBDT) 显式地构造了 Dirac、动力学 Dirac 和 Dirac-Weyl 系统的势、Darboux 矩阵函数和相应的解。通过这种方式,我们构建了具有无限潜力的非零系统的显式解决方案,包括阶梯状和幂律增长潜力。因此,构建的电位(系统)与针对平凡初始系统的 GBDT 积极研究的情况有根本的不同。本文的 GBDT 构造中使用了非平凡初始系统的广义矩阵特征值 A(不一定是对角线)和相应的广义矩阵特征函数 Π ( x )。这些 Π ( x ) 的显式表达式是新的,推导出这些表达式的方法可以应用于各种其他重要问题。对电子动力学和石墨烯理论感兴趣的 Dirac-Weyl 系统的情况进行了更详细的研究,并且在我们研究该系统的方法中出现了变量的广义分离。导出了 Weyl-Titchmarsh 函数的显式表达式(以伪实现的形式)。
更新日期:2021-02-01
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