We introduce a new method of proof for the essential self-adjointness of the Dirac Hamiltonian for a specific class of non-uniformly elliptic mixed initial-boundary value problems for the Dirac equation on smooth, asymptotically flat Lorentzian spacetimes admitting a Killing field that is timelike near and tangential to the boundary, combining results from the theory of symmetric hyperbolic systems with near-boundary elliptic methods. Our results apply in particular to the situation that the spacetime includes horizons, on which the Hamiltonian in general fails to be elliptic.

We present an integral spectral representation of the massive Dirac propagator in the non-extreme Kerr geometry in horizon-penetrating coordinates, which describes the dynamics of Dirac particles outside and across the event horizon, up to the Cauchy horizon. To this end, we define the Kerr geometry in the Newman–Penrose formalism by means of a regular Carter tetrad in advanced Eddington–Finkelstein-type coordi- nates and the massive Dirac equation in a chiral Newman–Penrose dyad representation in Hamiltonian form. After showing the essential self-adjointness of the Hamiltonian, we compute the resolvent of this operator 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. Then, by applying Stone’s formula to the spectral measure of the Hamiltonian, that is, by expressing the spectral measure in terms of this resolvent, we obtain an explicit integral representation of the Dirac propagator from its formal spectral decomposition.