Quantum theory on Lobatchevski spaces

In this paper, we set up a general formalism for dealing with quantum theories on a Lobatchevski space, i.e. a spatial manifold that is homogeneous, isotropic and has negative curvature. The heart of our approach is the construction of a suitable basis of plane waves which are eigenfunctions of the Laplace-Beltrami operator relative to the geometry of the curved space. These functions were previously introduced in the mathematical literature in the context of group theory; here we revisit and adapt the formalism in a way specific for quantum mechanics. Our developments render dealing with Lobatchevski spaces, which used to be quite difficult and a source of controversies, easily tractable. Applications to the Milne and de Sitter universes are discussed as examples