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

Application to anharmonic oscillators

Alexei V. Sergeev, David Z. Goodson

Research output: Contribution to journalArticle

57 Citations (Scopus)

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 languageEnglish
Pages (from-to)4301-4317
Number of pages17
JournalJournal of Physics A: Mathematical and General
Volume31
Issue number18
DOIs
Publication statusPublished - 8 May 1998
Externally publishedYes

Fingerprint

Algebraic function
Anharmonic Oscillator
Summation
Asymptotic Expansion
Branch
oscillators
perturbation
expansion
Perturbation Expansion
stairways
Parameter Perturbation
Recursive Algorithm
Quartic
Riemann Surface
Rayleigh
Algebraic Equation
Rate of Convergence
eigenvalues
Efficient Algorithms
Singularity

ASJC Scopus subject areas

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

Cite this

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

In: Journal of Physics A: Mathematical and General, Vol. 31, No. 18, 08.05.1998, p. 4301-4317.

Research output: Contribution to journalArticle

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