Complete families of solutions for the Dirac equation: an application of   bicomplex pseudoanalytic function theory and transmutation operators
  The Dirac equation with a scalar and an electromagnetic potentials is considered. In the time-harmonic case and when all the involved functions depend only on two spatial variables it reduces to a pair of decoupled bicomplex Vekua-type equations [8]. Using the technique developed for complex Vekua equations a system of exact solutions for the bicomplex equation is conctructed under additional conditions, in particular when the electromagnetic potential is absent and the scalar potential is a function of one Cartesian variable. Introducing a transmutation operator relating the involved bicomplex Vekua equation with the Cauchy-Riemann equation we prove the expansion and the Runge approximation theorems corresponding to the constructed family of solutions. 
Hugo M. Campos, Vladislav V. Kravchenko and Luis M. Mendez
math-ph math.AP math.CA math.CV math.MP