Analytic-continuation approach to the resummation of divergent series in Rayleigh-Schrödinger perturbation theory

Abstract

The method of analytic continuation is applied to estimate eigenvalues of linear operators from finite order results of perturbation theory even in cases when the latter is divergent. Given a finite number of terms ${E}^{(k)},k=1,2,\ensuremath{\cdots}M$ resulting from a Rayleigh-Schr\odinger perturbation calculation, scaling these numbers by ${\ensuremath{\mu}}^{k}$ ($\ensuremath{\mu}$ being the perturbation parameter) we form the sum $E(\ensuremath{\mu})={\ensuremath{\sum}}_{k}{\ensuremath{\mu}}^{k}{E}^{(k)}$ for small $\ensuremath{\mu}$ values for which the finite series is convergent to a certain numerical accuracy. Extrapolating the function $E(\ensuremath{\mu})$ to $\ensuremath{\mu}=1$ yields an estimation of the exact solution of the problem. For divergent series, this procedure may serve as resummation tool provided the perturbation problem has a nonzero radius of convergence. As illustrations, we treat the anharmonic (quartic) oscillator and an example from the many-electron correlation problem.