Quasi-bound states and late-time evolution of a massive fermion around a Reissner-Nordstr\"{o}m black hole

Abstract

A massive fermion around a charged black hole provides a gravitational analogue 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 quasi-bound-state spectrum, including fine-structure corrections and more accurate decay widths. The extremal Reissner-Nordstr\"{o}m case ($|Q|=M$) is treated separately and shown to be the smooth limiting result of the non-extremal 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 $1/m < t < 1/m^3M^2$. In the far late-time regime $t>1/m^3M^2$, the activation of the quasi-bound 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 quasi-bound contribution coexists with the familiar power-law component.