The Hamiltonian path integrals and the uniform semi-classical approximations for the propagator

The generalized path expansion scheme is defined for path integration in phase space. Within this framework the semi-classical limits to the propagator, both in the momentum and the coordinate representations are studied. It is shown that the role played by the Morse operator in the Lagrangian formulation of the path integral method is taken by another differential operator of the Dirac type. The relevant properties of this operator are discussed. The semi-classical approximations are obtained by extending the results of catastrophe theory for the asymptotic evaluation of finite-dimensional integrals to the domain of path integration. Various forms of the uniform semi-classical approximations are obtained. Their validity and applicability are discussed. The method is illustrated by a solution of a simple example in which non-generic catastrophe occurs. (author)