Numerical analysis of convergent perturbation expansions in quantum theory

We present the results of a numerical analysis of the convergence of the new perturbation expansion recently proposed by Belokurov, Solovyev, and Shavgulidze. Two particular examples are considered: the anharmonic oscillator in quantum mechanics and the renormalization group β-function in field theory. It is shown that in the first case, the series converges to an exact value in a wide range of expansion parameters. This range can be enlarged with the help of the Padé approximation. In field theory, the results have a stronger dependence on the regularization parameter. We discuss an algorithm for choosing this parameter that produces stable results. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 291–297, February, 1997.