The Quantum Development of an Asymptotically Euclidean Cauchy Hypersurface

In our model of quantum gravity the quantum development of a Cauchy hypersurface is governed by a hyperbolic equation in a fiber bundle derived as the result of a canonical quantization process. To find physically interesting solutions of the hyperbolic equation we employ the separation of variables by writing the solutions as products of temporal and spatial eigenfunctions or eigendistributions of self-adjoint differential operators defined in appropriate Hilbert spaces. There are differential operators which only act in the fibers and others which only act in the base space which is a Cauchy hypersurface of the quantized globally hyperbolic manifold. The operator acting in the base space is a Schrödinger operator and it should have a complete set of eigendistributions with positive eigenvalues. Assuming the Cauchy hypersurface to be asymptotically Euclidean we prove that this indeed the case and that eigendistributions are smooth tempered distributions. Together with the other eigenfunctions the existence of which are already proved they can be used to obtain the solutions of the hyperbolic equation.