We use the Fock-Ivanenko formalism to obtain the Dirac equation which describes the interaction of a massless 1 / 2 -spin neutral fermion with a gravitational field around a Schwarzschild black hole ( BH ) .
We obtain approximated analytical solutions for the eigenvalues of the energy ( quasi-normal frequencies ) and their corresponding eigenstates ( quasi-normal states ) .
The interesting result is that all the asymptotic states [ and their supersymmetric ( SUSY ) partners ] have a purely imaginary frequency , which can be expressed in terms of the Hawking temperature T _ { H } : E ^ { ( \uparrow \downarrow ) } _ { n } = -2 \pi i nT _ { H } .
Furthermore , as one expects for SUSY Hamiltonians , the isolated bottom state has a real null energy eigenvalue .