On the calculation of quantum mechanical ground states from classical geodesic motion on certain spaces of constant negative curvature

Abstract

We consider geodesic motion on three-dimensional Riemannian manifolds of constant negative curvature, topologically equivalent to S × ]0, 1[, S a compact surface of genus two. To those trajectories which are bounded and recurrent in both directions of the time evolution t → + ∞, t → − ∞ a fractal limit set is associated whose Hausdorff dimension is intimately connected with the quantum mechanical energy ground state, determined by the Schrödinger operator on the manifold. We give a rather detailed and pictorial description of the hyperbolic spaces we have in mind, discuss various aspects of classical and quantum mechanical motion on them as far as they are needed to establish the connection between energy ground state and Hausdorff dimension and give finally some examples of ground state calculations in terms of Hausdorff dimensions of limit sets of classical trajectories.