first order infinitesimal adjacent to the event horizon we can find the coefficients $a_1$ and $b_1$ by inserting it to the equation system $(\ref{76})$ and solving it. Because of the complexity of $a_1$ and $b_1$ we have effectively four equations. The calculation is quite long but mathematically trivial. We will therefore refrain from presenting it here.

The coefficients $a_1$ and $b_1$ of the first order series expansion Eq. $(\ref{77})$ together with the start values of $\phi$ and $\chi$ define the wave function infinitesimally close to the horizon from where the numerical integration can be started. The energy eigenvalues are found by the condition that the wave function has to approach zero at infinity to assure integrability.

Non-hermiticity of the Hamiltonian
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Unfortunately, we cannot be sure that the eigenvalues of the Hamiltonian in Eq. $(\ref{88})$ $H=H_{Dirac}+H_{Int}$ are real. In general real eigenvalues are only obtained from hermitean operators. In the case discussed here, we will show that the Hamiltonian is not hermitian. The Hamiltonian is the generator of the group describing the way quantum mechanical states in Hilbert space evolve with time. This means that

$$U(t_2,t_1)={\rm exp}\left[-iH(t_2-t_1)\right],
\label{91}$$

where $U(t_2,t_1)$ describes the relation between a quantum mechanical state $| \psi 