Evaluation of the density matrix for an ensemble of anharmonic oscillators by a path integral approach

Considering the Feynman path integral representation for the configuration-space density matrix for an ensemble of anharmonic oscillators, we determine the stationary paths near which the integrand remains stationary. By taking the path integral to be saturated by contributions from the neighborhood of the path which maximizes the integrand we evaluate the density matrix explicitly in analytic form. This seems to be the first such evaluation of a path integral for a system not describable by a quadratic Hamiltonian. We also comment briefly on the question of analyticity with respect to the perturbation parameter.