We calculate the bound-state energy spectrum of the Dirac Equation in a Schwarzschild black hole background using a minimax variational method. Our method extends that of Talman to the case of non-Hermitian interactions, such as a black hole. The trial function is expressed in terms of a basis set that takes into account both the Hermitian limit of the interaction in the non-relativistic approximation, and the general behaviour of the solutions at the origin, the horizon and infinity. Using this trial function an approximation to the full complex energy bound-state spectrum is computed. We study the behaviour of the method as the coupling constant of the interaction is increased, which increases both the relativistic effects and the size of the non-Hermitian part of the interaction. Finally we confirm that the method follows the expected Hylleraas-Undheim behaviour.
Alejandro Caceres and Chris Doran
Minimax determination of the energy spectrum of the Dirac equation in a Schwarzschild background
Phys. Rev. A 72, 022103 (2005)