Minimax determination of the energy spectrum of the Dirac equation in a Schwarzschild background

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 nonrelativistic approximation, and the general behavior 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 behavior 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 behavior.

Figure 1

Rayleigh quotient as function of the nonlinear parameters $a,b$. $\alpha =0.1,k=-1$; minimax at $a=8.9186905\times {10}^{-2},b=7.8238289\times {10}^{-2}$.

Figure 2

Convergence to a single value of the approximation to the real part of the energy $\mathrm{Re}\left(E\right)$. For values $\alpha =0.1,k=-1$, the energy $E∕\left(m{c}^2\right)=0.9947208-i2.0498160\times {10}^{-5}$ is obtained.

Figure 3

Real part of the energy as a function of the black-hole mass. The real part of the energy given by the minimax (broken line) and by the shooting energy (solid line) are compared for the ground state with $k=-1$.

Figure 4

Decay rate as a function of the black-hole mass. The decay rates given by the minimax (broken line) and the shooting (solid line) corresponding to Fig. 3.

Figure 5

Hylleraas-Undheim behavior of the positive real part of the eigenenergies, given the values $\alpha =0.1,k=-1$, and the trial function (30).

Figure 6

Hylleraas-Undheim of the negative real part of the eigenenergies, given the values $\alpha =0.1,k=-1$, and the trial function (30).