Relation between full traces of Green functions for initial and Darboux transformed Dirac problems

TL;DR

This paper explores the connection between Green functions of original and Darboux-transformed Dirac problems, validating the completeness of transformed eigenfunctions under general potentials and boundary conditions.

Contribution

It establishes a general relation between Green functions of initial and Darboux-transformed Dirac problems, aiding in the analysis of their spectral properties.

Findings

Derived a relation between Green functions for transformed and initial problems

Confirmed completeness of eigenfunctions after Darboux transformation

Applicable to Dirac problems with general potentials and boundary conditions

Abstract

We establish the relation between full traces of the Green functions for some initial and the Darboux transformed one-dimensional two component Dirac problems with the most general form of potential. The result is used to check the completeness of set of wave functions obtained by the Darboux transformation of the eigenfunctions set for the initial Dirac problem with some typical boundary conditions.