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

,

into a convergent series in powers

,

where 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 . 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 . 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.

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