A formal approach to weak and strong coupling expansions and its application to the quantum anharmonic oscillator

The main issue of this work consists in approximating a physical quantity from its perturbative expansion. The present extension method puts a given p-order weak coupling expansion of a physical quantity in a one-to-one relationship with a nonperturbative expression. This expression serves as a p-order estimate of the physical quantity for all values of the coupling constant. The method goes through the introduction of a variational parameter, and an optimization procedure. It is original in the way it simply takes into account the known “form” of the strong coupling expansion together with the known coefficients of the weak coupling expansion of the searched quantity. Applying it to the study of the quantum anharmonic oscillator in one dimension provides high degree of accuracy from lower orders of perturbation.