  We consider the massive Dirac equation in the non-extreme Kerr geometry in
horizon-penetrating advanced Eddington-Finkelstein-type coordinates and derive
a functional analytic integral representation of the associated propagator
using the spectral theorem for unbounded self-adjoint operators, Stone's
formula, and quantities arising in the analysis of Chandrasekhar's separation
of variables. This integral representation describes the dynamics of Dirac
particles outside and across the event horizon, up to the Cauchy horizon. In
the derivation, we first write the Dirac equation in Hamiltonian form and show
the essential self-adjointness of the Hamiltonian. For the latter purpose, as
the Dirac Hamiltonian fails to be elliptic at the event and the Cauchy horizon,
we cannot use standard elliptic methods of proof. Instead, we employ a new,
general method for mixed initial-boundary value problems that combines results
from the theory of symmetric hyperbolic systems with near-boundary elliptic
methods. In this regard and since the time evolution may not be unitary because
of Dirac particles impinging on the ring singularity, we also impose a suitable
Dirichlet-type boundary condition on a time-like inner hypersurface placed
inside the Cauchy horizon, which has no effect on the dynamics outside the
Cauchy horizon. We then compute the resolvent of the Dirac Hamiltonian via the
projector onto a finite-dimensional, invariant spectral eigenspace of the
angular operator and the radial Green's matrix stemming from Chandrasekhar's
separation of variables. Applying Stone's formula to the spectral measure of
the Hamiltonian in the spectral decomposition of the Dirac propagator, that is,
by expressing the spectral measure in terms of this resolvent, we obtain an
explicit integral representation of the propagator.
