Darboux Transformation of the Green Function for the Dirac Equation with the Generalized Potential

TL;DR

This paper develops a method to transform Green functions of the one-dimensional Dirac equation using Darboux transformations, providing explicit formulas and illustrating with free particle examples.

Contribution

It introduces a new approach to obtain Green functions after Darboux transformations for the Dirac equation with boundary conditions.

Findings

Derived explicit Green functions for transformed Dirac equations.

Established a formula for the trace difference of Green functions.

Applied the method to free particle Dirac equations on an interval.

Abstract

We consider the Darboux transformation of the Green functions of the regular boundary problem of the one-dimensional stationary Dirac equation. We obtained the Green functions of the transformed Dirac equation with the initial regular boundary conditions. We also construct the formula for the unabridged trace of the difference of the transformed and the initial Green functions of the regular boundary problem of the one-dimensional stationary Dirac equation. We illustrate our findings by the consideration of the Darboux transformation for the Green function of the free particle Dirac equation on an interval.