Jonathan Blackledge and Bazar Babajanov
We consider a method of solving the Dirac scattering problem based on an approach previously used by the authors to solve the Schr¨odinger scattering problem to develop a conditional exact scattering solution and an unconditional series solution. We transform the Dirac scattering problem into a form that facilitates a solution based on the relativistic Lippmann-Schwinger equation using the relativistic Green’s function that is transcendental in terms of the scattered field. Using the Dirac operator, this solution is transformed further to yield a modified relativistic Lippmann-Schwinger equation that is also transcendental in terms of the scattered field. This modified solution facilitates a condition under which the solution for the scattered field is exact. Further, by exploiting the simultaneity of the two solutions available, we show that is possible to define an exact (non-conditional) series solution to the problem.