Exact evolution operator on non-compact group manifolds

Free quantal motion on group manifolds is considered. The Hamiltonian is given by the Laplace – Beltrami operator on the group manifold, and the purpose is to get the (Feynman’s) evolution kernel Kt. The spectral expansion, which produced a series of the representation characters for Kt in the compact case, does not exist for non-compact group, where the spectrum is not bounded. In this work real analytical groups are investigated, some of which are of interest for physics. An integral representation for Kt is obtained in terms of the Green function, i.e. the solution to the Helmholz equation on the group manifold. The alternative series expressions for the evolution operator are reconstructed from the same integral representation, the spectral expansion (when exists) and the sum over classical paths. For non-compact groups, the latter can be interpreted as the (exact) semiclassical approximation, like in the compact case. The explicit form of Kt is obtained for a number of non-compact groups.