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

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 $e^{-const.\times N^{1-\omega }}$. We present a review of the method and various applications.

32 pages

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