2011/11/17 by Campos, Hugo M., Kravchenko, Vladislav V., Mendez, Luis M.
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1111.4198
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.