A unified formal approach to weak and strong coupling expansions

TL;DR

This paper introduces a systematic, unified formal method for summing divergent perturbation series in physical systems with two parameters, applicable to weak and strong coupling regimes, demonstrated on quantum anharmonic oscillators.

Contribution

It presents a novel extension procedure that unifies weak and strong coupling expansions, providing accurate estimates of divergent series sums with good convergence properties.

Findings

Accurately estimates sums of divergent perturbation series.

Applicable to quantum anharmonic oscillators in various dimensions.

Shows high accuracy from low-order perturbation terms.

Abstract

The main issue of this work consists in extracting one or several finite values for the sum of series involved in perturbation theories. It is supposed to work for all cases in which two physical parameters are involved, and makes thorough use of dimensional arguments concerning these physical quantities. Weak and strong coupling expansions are considered as the two faces of a formal expression which is the central object of this method. This so-called extension procedure is systematic. We apply it here to the divergent perturbative expansion of the ground state energy of the anharmonic oscillator of quantum mechanics in zero and one dimension, and provide, given a p-order divergent expansion, an analytical expression of its estimated sum. This method which is inspired by variational procedures provides high degree of accuracy from lower orders of perturbation and seems to have remarkable converge properties in a wide class of expansions concerning physical observables.