into a system of two differential equations we obtain

$$\begin{aligned}
\left[-\partial_r+\frac{w(J+\frac{1}{2})}{r}\right]\Phi &=\left[E-m-i\left(\frac{2M}{r}\right)^\frac{1}{2}(\partial_r-\frac{1}{4r})\right]\chi, \nonumber\\
\left[\partial_r+\frac{w(J+\frac{1}{2})}{r}\right]\chi 
&= \left[E+m-i\left(\frac{2M}{r}\right)^\frac{1}{2}(\partial_r-\frac{1}{4r})\right]\Phi.
\label{75}\end{aligned}$$

These are the differential equations for a Dirac particle in a Schwarzschild metric we have to solve.

Solution of the time independent equation with suitable boundary conditions
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After deriving the system of differential equations the final task is to calculate the energy eigenstates by solving the set of coupled partial differential equations. Because the set of equations contains complex numbers it is convenient to split the two radial components $\Phi$ and $\chi$ in a real and imaginary component resulting in four equations which can be integrated numerically.

However, before we do this, we have to make sure that there exists a regular solution at the event horizon and to determine the necessary boundary conditions. Therefore we will transpose the above equation system. In both Eqs. $(\ref{75})$ appear the derivatives of $\Phi$ and $\chi$. By solving one equation for $\partial_r \Phi$ and the other for $\partial_r \chi$ and inserting each equation into the other we get the following system

$$\begin{aligned}
\left(1-\frac{2M}{r}\right)\partial_r\Phi&=\left[\frac{w(J+\frac{1}{2})}{r}+i\left(\frac{2M}{r}\right)^\frac{1}{2}[E+m+i\left(\frac{2M}{r}\right)^\frac{1}{2}\frac{1}{4r}]\right]\Phi\nonumber\\
&\quad+\left[i\left(\frac{2M}{r}\right)^\frac{1}{2}[-\frac{w(J+\frac{1}{2})}{r}-\frac{1}{4r}]-(E-m)\right]\chi,\nonumber\\
\left(1-\frac{2M}{r}\right)\partial_r\chi&=\left[-\frac{w(J+\frac{1}{2})}{r}+i\left(\frac{2M}{r}\right)^\frac{1}{2}[E-m+i\left(\frac{2M}{r}\right)^\frac{1}{2}\frac{1}{4r}]\right]\chi\nonumber\\
&\quad+\left[i\left(\frac{2M}{r}\right)^\frac{1}{2}[-\frac{w(J+\frac{1}{2})}{r}+\frac{1}{4r}]+(E+m)\right]\Phi.
\label{76}\end{aligned}$$

After this transformation the singularity at the event horizon $(r_s=2M)$ shows up explicitly. Nevertheless it is possible to 