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

卷:275

Article

Sakhnovich, Alexander¹ 

[1] Univ Vienna, Fac Math, Oskar Morgenstern Pl 1, A-1090 Vienna, Austria

关键词: Dirac system;    Dynamical Dirac system;    Dirac-Weyl system;    Explicit solution;    Electron dynamics;    Generalised matrix eigenvalue;   

DOI  :  10.1016/j.jde.2020.11.037

来源: Elsevier

【 摘 要 】

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 Pi(x) of the nontrivial initial systems are used in the GBDT constructions in this paper. Explicit expressions for these Pi(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. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free