Strong-coupling limits induced by weak-coupling expansions

Abstract

A method is described for the extrapolation of perturbative expansions in powers of asymptotically small coupling parameters or other variables onto the region of finite variables and even to the variables tending to infinity. The method involves the combination of ideas from renormalization group theory, approximation theory, dynamical theory, and optimal control theory. The extrapolation is realized by means of self-similar factor approximants, whose control parameters can be uniquely defined. The method allows to find the large-variable behavior of sought functions knowing only their small-variable expansions. Convergence and accuracy of the method are illustrated by explicit examples, including the so-called zero-dimensional field theory and anharmonic oscillator. Strong-coupling behavior of Gell-Mann-Low functions in multicomponent field theory, quantum electrodynamics, and quantum chromodynamics is found, being based on their weak-coupling perturbative expansions.