Black holes have long served as a testing ground for probing theories of gravity and quantum mechanics. Notably, fundamental fields in the neighborhood of black holes exhibit rich phenomena that could yield astrophysically observable signatures. However, exploring these structures typically requires computationally intensive numerical calculations. In this work, the dynamics of a massive Dirac field outside a Schwarzschild black hole are revisited. We propose a novel matching scheme that enables the analytical solution of the coupled first-order Dirac equation, as opposed to the conventional second-order approach. This method yields a compact and unified analytical expression for the energy spectrum, which shows improved agreement with the numerical results. The improvement comes from the higher-order correction of the angular parameter that was ignored previously.

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