On the Method of Stationary Phase in Calculating the Propagator of the Coulomb–Kepler Problem

Abstract

A theorem is proven on the method of stationary phase to be applied in the calculation of the propagator from the Green’s function of the Coulomb - Kepler problem, by means of Fourier transformation. The Green’s function is available in position representation in an exact compact form as ingeniously derived by Hostler (1963). The Fourier transform, at first, exists in the distributional sense. In order to avoid series representation in Schwartz space, the integrals are defined away from the real axis of the integration variable and by taking principal values. The theorem deals with integrands with a highly oscillating exponential term. It is shown that the method of stationary phase renders exact results in cases relevant to the problem, whereby the integrand can contain an analytic amplitude factor in addition to the exponential term.