find a regular solution. To obtain this solution we perform a series expansion around the event horizon

$$\Phi=\sum_{i=1}^{\infty} a_i(r-2M)^i\quad,\quad\chi=\sum_{i=1}^\infty b_i(r-2M)^i.
\label{77}$$

By inserting now the series expansion into Eq. $(\ref{76})$ we see that it is a solution of our differential equation system at the event horizon $r=2M$ under condition that

$$\begin{aligned}
\left[iE-\frac{1}{8M}+\left(\frac{\omega(J+\frac{1}{2})}{2M}+im\right)\right]a_0&\nonumber\\
\quad=\left[E+i\frac{1}{8M}+\left(i\frac{\omega(J+\frac{1}{2})}{2M}-m\right)\right]b_0&.
\label{78}
$$

Here we have used $$\sum_{i=1}^{\infty} x_i(r-2M)^i\vert_{r=2M}=x_0,
\label{79}$$ and $$\begin{aligned}
\left(1-\frac{2M}{r}\right)\partial_r\sum_{i=1}^{\infty} x_i(r-2M)^i\vert_{r=2M}&\nonumber\\
\quad=\frac{r-2M}{r}\sum_{i=1}^{\infty} ix_i(r-2M)^{(i-1)}\vert_{r=2M}\nonumber\\ 
\quad=\frac{1}{r}\sum_{i=1}^{\infty} ix_i(r-2M)^i\vert_{r=2M}=0.
\label{80}\end{aligned}$$ To begin an integration of the differential equation system Eq. $(\ref{76})$ one needs the start values of $\Phi$ and $\chi$. By using the series expansion Eq. $(\ref{77})$ we have found one constraint for the start values of the integration of $\phi$ and $\chi$. The remaining degree of freedom is only connected to the the overall normalisation of the wave function. It is determined by the normalisation condition that the integral over the squared norm has to be equal to unity because of the probability interpretation of the wave function. So the wave is defined apart from a physically irrelevant overall phase.

The integration cannot be started directly from the horizon because of the problem with the non-regular solutions. However, if we consider our series expansion Eq. $(\ref{77})$ to the 