# Variational perturbation theory

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In mathematics, variational perturbation theory (VPT) is a mathematical method to convert divergent power series in a small expansion parameter, say

$s=\sum_{n=0}^\infty a_n g^n$,

into a convergent series in powers

$s=\sum_{n=0}^\infty b_n /(g^\omega)^n$,

where $\omega$ is a critical exponent (the so-called index of "approach to scaling" introduced by Franz Wegner). This is possible with the help of variational parameters, which are determined by optimization order by order in $g$. The partial sums are converted to convergent partial sums by a method developed in 1992.[1]

Most perturbation expansions in quantum mechanics are divergent for any small coupling strength $g$. They can be made convergent by VPT (for details see the first textbook cited below). The convergence is exponentially fast.[2][3]

After its success in quantum mechanics, VPT has been developed further to become an important mathematical tool in quantum field theory with its anomalous dimensions.[4] Applications focus on the theory of critical phenomena. It has led to the most accurate predictions of critical exponents. More details can be read here.

## References

1. ^
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3. ^ Guida, R.; Konishi, K.; Suzuki, H. (1996). "Systematic Corrections to Variational Calculation of Effective Classical Potential". Annals of Physics 249 (1): 109–145. arXiv:hep-th/9505084. Bibcode:1996AnPhy.249..109G. doi:10.1006/aphy.1996.0066.
4. ^