A massive fermion around a charged black hole provides a gravitational analog of atomic bound states and their relaxation. In this work, we study this system by formulating the radial equation as a coupled matrix system and constructing the Green’s function with ingoing boundary conditions at the horizon and decaying boundary conditions at infinity. In the weak-coupling scenario $|qQ|\sim mM<1$, a matrix matching scheme gives an improved analytic expression of quasibound-state spectrum, including fine-structure corrections and more accurate decay widths. The extremal Reissner-Nordström case ($|Q|=M$) is treated separately and shown to be the smooth limiting result of the nonextremal spectrum. We further analyze the branch-cut contribution to the time-domain Green’s function in the late-time limit. We confirm an oscillatory power-law behavior in intermediate late-time regime $1/m<t<1/m^3{M}^2$. In the far late-time regime $t>1/m^3{M}^2$, the activation of the quasibound states produces an $t^{-5/6}\exp (-\eta t^{1/3})$ suppression with a chirping phase before the asymptotic $t^{-5/6}$ tail previously found in the limit $t\to \infty$. Direct time-domain simulations support this distinction and show how the quasibound contribution coexists with the familiar power-law component.

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• Classical black holes
• Dirac equation
• Fluids & classical fields in curved spacetime
• Quantum fields in curved spacetime