# A New Method for Obtaining Solutions of the Dirac Equation

### Vladislav V. Kravchenko

Cinvestav del IPN, Santiago De Querétaro, Mexico

## Abstract

The Dirac operator with pseudoscalar, scalar or electric potential and the Schrödinger operator are considered. For any potential depending on an arbitrary function $ξ$ satisfying the equation

where $γ(ξ)=∣gradξ∣$ there are constructed special solutions of the Dirac and the Schrödinger equations, and in some cases the fundamental solutions are obtained also. The class of solutions of equation ($∗$) is sufficiently ample. For example, if 1) $ξ$ is harmonic and 2) the gradient squared of $ξ$ is constant, then $ξ$ satisfies (*). That is, in particular, any complex linear combination of three variables$ξ=ax_{1}+bx_{2}+cx_{3}+d$ satisfies equation $(∗)$, and the solutions may be obtained for any potential depending on such $ξ$. All results are obtained using some special biquaternionic projection operators constructed after having solved an eikonal equation corresponding to $ξ$.

## Cite this article

Vladislav V. Kravchenko, A New Method for Obtaining Solutions of the Dirac Equation. Z. Anal. Anwend. 19 (2000), no. 3, pp. 655–676

DOI 10.4171/ZAA/973