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Zeitschrift für Analysis und ihre Anwendungen

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Volume 19, Issue 3, 2000, pp. 655–676
DOI: 10.4171/ZAA/973

Published online: 2000-09-30

A New Method for Obtaining Solutions of the Dirac Equation

Vladislav V. Kravchenko[1]

(1) Cinvestav del IPN, Santiago De Querétaro, Mexico

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 $$\Delta \xi – \gamma (\xi) \cdot \frac{d \gamma (\xi)}{d \xi} = 0$$ 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$\xi = ax_1 + bx_2 + cx_3 + d$ 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 $\xi$.

Keywords: Dirac equation, quaternionic analysis

Kravchenko Vladislav: A New Method for Obtaining Solutions of the Dirac Equation. Z. Anal. Anwend. 19 (2000), 655-676. doi: 10.4171/ZAA/973