A CONNECTED-GRAPH EXPANSION OF THE ANHARMONIC-OSCILLATOR PROPAGATOR

Two series representations of the propagator associated with the quantum anharmonic-oscillator are developed. A closed form for the Dyson series expansion of the propagator is obtained by using a phase-space perturbation technique. For Abelian interactions the Dyson series can be rearranged into an exponentiated connected-graph series. This representation is structurally similar to the cluster expansions of propagators associated with perturbations of the kinetic-energy Hamiltonian. The connected-graph series is particularly useful for the development of semiclassical expansions such as the WKB expansion. The analytic structure in the physical parameters such as mass, time and Planck's constant is readily extracted from the connected-graph series because of the explicit closed-form formula for the series.