within the horizon it cannot escape and has to end at the singularity in the centre. Without this special structure of space time connected to black holes such a strange phenomena could not take place. This means that the decay rate of the states depends on the properties of the wave function near the origin. A small probability density near the singularity leads to a long decay time. Thus states with a large angular momentum, i.e. a small probability density near the origin are very stable. This shows up explicitly in the decreasing imaginary part of the energy as a function of the displacement of the probability density outwards (see Figures). For a more extensive discussion of this decay phenomenon the reader can consult ref. [@Lasenby:2002mc].

Figures and Discussion
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The solutions of the Dirac equation in a Schwarzschild background differ significant from the solutions of the corresponding equation with a coupled Coulomb-potential describing a hydrogen atom. The main difference is the non-stability of the states connected to the non-Hermiticity of the Hamiltonian. In the gravitational field real bound states exist only in an approximative sense and if the angular momentum becomes large the main part of the probability density 