Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Venkat Abhignan

TL;DR

This work tackles the challenge of divergent perturbation series in quantum atomic problems by applying a parameter-free continued exponential resummation, drawing parallels to Padé-based convergence. By transforming perturbation expansions for He-like ground-state energies, Hydrogen Stark, and Zeeman effects into continued-exponential forms and extracting coefficients from low-order perturbative data (up to $i=9$ or $i=16$ in some cases), the authors achieve rapid convergence and, in many instances, tight bounds or high-precision eigenvalues. They demonstrate performance improvements via Shanks transformations and validate results against established literature, while highlighting the method’s robustness across different perturbation regimes and its placement within a broader resummation framework. The approach offers a simple, parameter-free alternative to Padé approximants for turning divergent series into accurate analytic approximations with potential applicability to other perturbative contexts.

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.

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

TL;DR

This work tackles the challenge of divergent perturbation series in quantum atomic problems by applying a parameter-free continued exponential resummation, drawing parallels to Padé-based convergence. By transforming perturbation expansions for He-like ground-state energies, Hydrogen Stark, and Zeeman effects into continued-exponential forms and extracting coefficients from low-order perturbative data (up to or in some cases), the authors achieve rapid convergence and, in many instances, tight bounds or high-precision eigenvalues. They demonstrate performance improvements via Shanks transformations and validate results against established literature, while highlighting the method’s robustness across different perturbation regimes and its placement within a broader resummation framework. The approach offers a simple, parameter-free alternative to Padé approximants for turning divergent series into accurate analytic approximations with potential applicability to other perturbative contexts.

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.
Paper Structure (6 sections, 12 equations, 1 figure, 9 tables)

This paper contains 6 sections, 12 equations, 1 figure, 9 tables.

Table of Contents

  1. Introduction
  2. Ground-state energy of He isoelectronic species
  3. Stark effect in H atom
  4. H atom with radial perturbation
  5. Zeeman effect in H atom
  6. Conclusion

Figures (1)

  • Figure 1: Estimates of ground-state energy of He atom from continued exponential $A_i$ and its Shanks transforms $S(A_i), S^2(A_i)$ at successive orders compared with result obtained by Hylleraas-Knight-Scherr variational perturbation method PhysRevA.41.1247 (horizontal green line). The lower image at the smaller scale shows that $A_i$ has convergence, while its Shanks transforms $S(A_i), S^2(A_i)$ show rapid convergence towards the exact value.