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

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 Padé 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.

References in corpus (6)

• Resummation of the Divergent Perturbation Series for a Hydrogen Atom in an Electric Field
• Continued functions and perturbation series: Simple tools for convergence of diverging series in $O(n)$-symmetric ${\varphi }^4$ field theory at weak coupling limit
• Extrapolation from hypergeometric functions, continued functions and Borel-Leroy transformation; Resummation of perturbative renormalization functions from field theories
• High orders of the perturbation theory for hydrogen atom in magnetic field
• Continued functions and Borel-Leroy transformation: Resummation of six-loop ε-expansions from different universality classes
• Continued functions and critical exponents: Tools for analytical continuation of divergent expressions in phase transition studies