On the classes of explicit solutions of Dirac, dynamical Dirac and Dirac–Weyl systems with non-vanishing at infinity potentials, their properties and applications

Abstract

We construct explicitly potentials, Darboux matrix functions and corresponding solutions of Dirac, dynamical Dirac and Dirac–Weyl systems using generalised Bäcklund-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.