The spectral theory of Laplace(-Beltrami) and Dirac operators is analyzed in a global Lorentzian geometric setting. The two types of Lorentzian manifolds to be considered are 1) Time dependent, globally hyperbolic manifolds and 2) Static or stationary, non-globally hyperbolic manifolds. Self adjoint extensions of the above operators are studied. In addition to proving the existence of self adjoint extensions of those operators, we study the uniqueness of those extensions by analyzing essential self-adjointness. Applications to General Relativity and quantum field theory in curved spacetimes are worked out.
The main method for the wave equation is to apply recent results by Shubin showing that on a complete Riemannian manifold, the weighted Laplace-Beltrami operator plus a locally square integrable potential is essentially self-adjoint on the space of smooth functions of compact support. Preliminary works show that a sufficient condition for essential self-adjointness is that, after a suitable conformal transformation, the induced Riemannian metric is geodesically complete on each leaf of the foliation. Consequently, the second part of this project is to classify the time-dependent globally hyperbolic manifolds for which this condition can be satisfied. The third part of the project is to extend these methods to the study of essential self-adjointness of the Laplace and Dirac operators in static or stationary, non globally hyperbolic manifolds. In the last part we use these results to construct complex structures on the solution spaces as needed for the quantization.