Convergence in divergent series related to perturbation methods using continued exponential and Shanks transformations

TL;DR

This paper explores the use of continued exponential and Shanks transformations to achieve convergence in divergent perturbation series for quantum energy calculations, demonstrating similarities to Padé approximation without free parameters.

Contribution

It introduces a novel application of continued exponential and Shanks transformations to improve convergence of divergent series in quantum perturbation theory.

Findings

Convergence properties similar to Padé approximation are observed.

Effective convergence achieved using only first few terms of series.

No free parameters are needed for the convergence process.

Abstract

Divergent solutions are ubiquitous with perturbation methods. We use continued function such as continued exponential to converge divergent series in perturbation approaches for energy eigenvalues of Helium, Stark effect and Zeeman effect on Hydrogen. We observe that convergence properties are obtained similar to that of the Pad\'e approximation which is extensively used in literature. Free parameters are not used which influence the convergence and only first few terms in the perturbation series are implemented.