A New Method for Obtaining Solutions of the Dirac Equation

Abstract

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

where $\gamma (\xi )=|\mathrm{grad}\xi |$ 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 ($\ast$) is sufficiently ample. For example, if 1) $\xi$ is harmonic and 2) the gradient squared of $\xi$ is constant, then $\xi$ satisfies (*). That is, in particular, any complex linear combination of three variables$\xi =a{x}_1+b{x}_2+c{x}_3+d$ satisfies equation $(\ast )$, and the solutions may be obtained for any potential depending on such $\xi$. All results are obtained using some special biquaternionic projection operators constructed after having solved an eikonal equation corresponding to $\xi$.