### Abstract

The divergent Rayleigh-Schrödinger perturbation expansions for energy eigenvalues of cubic, quartic, sextic and octic oscillators are summed using algebraic approximants. These approximants are generalized Padé approximants that are obtained from an algebraic equation of arbitrary degree. Numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree. Different branches of the approximants converge to different branches of the function. The success of the high-degree approximants is attributed to their ability to model the function on multiple sheets of the Riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter. An efficient recursive algorithm for computing the diagonal approximant sequence is presented.

Original language | English |
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Pages (from-to) | 4301-4317 |

Number of pages | 17 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 31 |

Issue number | 18 |

DOIs | |

Publication status | Published - 8 May 1998 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Physics A: Mathematical and General*,

*31*(18), 4301-4317. https://doi.org/10.1088/0305-4470/31/18/018

**Summation of asymptotic expansions of multiple-valued functions using algebraic approximants : Application to anharmonic oscillators.** / Sergeev, Alexei V.; Goodson, David Z.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and General*, vol. 31, no. 18, pp. 4301-4317. https://doi.org/10.1088/0305-4470/31/18/018

}

TY - JOUR

T1 - Summation of asymptotic expansions of multiple-valued functions using algebraic approximants

T2 - Application to anharmonic oscillators

AU - Sergeev, Alexei V.

AU - Goodson, David Z.

PY - 1998/5/8

Y1 - 1998/5/8

N2 - The divergent Rayleigh-Schrödinger perturbation expansions for energy eigenvalues of cubic, quartic, sextic and octic oscillators are summed using algebraic approximants. These approximants are generalized Padé approximants that are obtained from an algebraic equation of arbitrary degree. Numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree. Different branches of the approximants converge to different branches of the function. The success of the high-degree approximants is attributed to their ability to model the function on multiple sheets of the Riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter. An efficient recursive algorithm for computing the diagonal approximant sequence is presented.

AB - The divergent Rayleigh-Schrödinger perturbation expansions for energy eigenvalues of cubic, quartic, sextic and octic oscillators are summed using algebraic approximants. These approximants are generalized Padé approximants that are obtained from an algebraic equation of arbitrary degree. Numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree. Different branches of the approximants converge to different branches of the function. The success of the high-degree approximants is attributed to their ability to model the function on multiple sheets of the Riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter. An efficient recursive algorithm for computing the diagonal approximant sequence is presented.

UR - http://www.scopus.com/inward/record.url?scp=0032495961&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032495961&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/31/18/018

DO - 10.1088/0305-4470/31/18/018

M3 - Article

VL - 31

SP - 4301

EP - 4317

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 18

ER -