  We formulate the Dirac equation for a massive neutral spin-half particle on a
rotating black hole spacetime, and we consider its (quasi)bound states:
gravitationally-trapped modes which are regular across the future event
horizon. These bound states decay with time, due to the absence of
superradiance in the (single-particle) Dirac field. We introduce a practical
method for computing the spectrum of energy levels and decay rates, and we
compare our numerical results with known asymptotic results in the small-$M
\mu$ and large-$M \mu$ regimes. By applying perturbation theory in a
horizon-penetrating coordinate system, we compute the `fine structure' of the
energy spectrum, and show good agreement with numerical results. We obtain data
for a hyperfine splitting due to black hole rotation. We evolve generic initial
data in the time domain, and show how Dirac bound states appear as spectral
lines in the power spectra. In the rapidly-rotating regime, we find that the
decay of low-frequency co-rotating modes is suppressed in the (bosonic)
superradiant regime. We conclude with a discussion of physical implications and
avenues for further work.
