Large-order estimates for ground-state energy perturbation series

A general treatment is given, using path integral methods, of obtaining accurate estimates on the rate of growth at large order of the perturbation coefficients for the lowest eigenvalue (ground-state energy) of a large class of anharmonic oscillators. Simple sufficient conditions are given on the potentialV(x) so that accurate upper and lower bounds on the perturbation coefficients may be derived. Several examples are given which generalize previous results. Examples from Euclidean quantum field theory are also considered.This paper is dedicated to the memory of Mark Kac.