Self-adjointness of the Dirac Hamiltonian for a class of non-uniformly elliptic boundary value problems

Abstract

We consider a boundary value problem for the Dirac equation in a smooth, asymptotically flat Lorentzian manifold admitting a Killing field which is timelike near and tangential to the boundary. A self-adjoint extension of the Dirac Hamiltonian is constructed. Our results also apply to the situation that the spacetime includes horizons, where the Hamiltonian fails to be elliptic.

An integral spectral representation of the massive Dirac propagator in the Kerr geometry in Eddington–Finkelstein-type coordinates

“…In this section, we show that the Dirac Hamiltonian in the non-extreme Kerr geometry in horizonpenetrating advanced Eddington-Finkelstein-type coordinates is essentially self-adjoint using the results obtained in [15] (we recently learned that in [19] related results were found with different methods).…”

“…Therefore, we cannot employ standard techniques and results from elliptic theory in order to verify the essential self-adjointness of the Dirac Hamiltonian. Instead, we apply the results derived in [15], where near-boundary elliptic methods are combined with results from the theory of symmetric hyperbolic systems (see, e.g., [2,6,16,23]). In the following, we state the geometrical and functional analytic settings for the formulation of the Cauchy problem for the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating coordinates in Hamiltonian form, which is used as a technical tool in the proof of the essential self-adjointness of the Dirac Hamiltonian.…”

“…where ∂ τ and ∂ φ are the Killing fields describing the stationarity and axisymmetry of the Kerr geometry, respectively. We note that the specific proof of existence of unique, global solutions of the Cauchy problem for the massive Dirac equation in Hamiltonian form for a general class of mixed initial-boundary value problems on Lorentzian manifolds presented in [15] makes essential use of a Killing field K = ∂ t that is tangential to and time-like on the inner boundary ∂M and represented by a coordinate system describing an observer who is co-moving along the associated flow lines. This Killing field may also be space-like or null in M \∂M .…”

“…This boundary condition prevents Dirac particles from impinging on the curvature singularity without affecting their dynamics in the region outside the Cauchy horizon, consequently leading to a unitary time evolution. Then, we apply the method of proof for the essential self-adjointness of the Dirac Hamiltonian for mixed initial-boundary value problems that are not uniformly elliptic introduced in [15]. Subsequently, it is possible to derive an integral representation of the Dirac propagator via the spectral theorem for unbounded self-adjoint operators…”

Lectures on Linear Stability of Rotating Black Holes

“…More details on the Dirac equation and the above method can be found in my joint papers with Niky Kamran, Joel Smoller and Shing-Tung Yau [14,16,15]. For the general method of constructing self-adjoint extensions, the more recent paper [22] may be useful.…”