Converting Divergent Weak-Coupling into Exponentially Fast Convergent Strong-Coupling Expansions

TL;DR

This paper introduces a variational method to transform divergent weak-coupling series into exponentially fast converging strong-coupling expansions, enabling accurate approximations outside the original convergence radius.

Contribution

It presents a novel variational approach that converts divergent series into rapidly converging strong-coupling expansions with exponential error decay.

Findings

Method achieves exponential convergence for large N.

Applicable to various divergent series in physics.

Improves accuracy of strong-coupling approximations.

Abstract

With the help of a simple variational procedure it is possible to convert the partial sums of order $N$ of many divergent series expansions $f(g)=\sum_{n=0}^\infty a_n g^n$ into partial sums $\sum_{n=0}^N b_n g^{- \omega n}$, where $0<\omega<1$ is a parameter that parametrizes the approach to the large-$g$ limit. The latter are partial sums of a strong-coupling expansion of $f(g)$ which converge against $f(g)$ for $g$ {\em outside} a certain divergence radius. The error decreases exponentially fast for large $N$, like $e^{-{\rm const.}\times N^{1-\omega}}$. We present a review of the method and various applications.