On the summation of divergent perturbation series in quantum mechanics and field theory

The possibility of recovering the Gell-Mann–Low function in the asymptotic strong-coupling regime by known first-order perturbation-theory (PT) terms βn and their asymptotics as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\tilde \beta _n $$ \end{document} as n → ∞ is investigated. Conditions are formulated that are necessary for recovering the required function at the physical level of rigor: (1) a large number of PT coefficients are known whose asymptotics has already been established, and (2) there is no intermediate asymptotics. Higher orders of PT, their asymptotic behavior, and power corrections are calculated in quantum mechanical problems that involve divergent PT series (including series for a funnel potential, the ϕ(0)4 model, and the Stark effect in a strong field). The scalar field theory ϕ(4)4 is considered in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overline {MS} $$ \end{document} and MOM regularization schemes. It is shown that one cannot make any definite conclusion about the asymptotics of the Gell-Mann–Low function as g → ∞ on the basis of information available for the above theory.