Summation of Rayleigh - Schrödinger perturbation series by algebraic Padé approximants: anharmonic oscillators

Alexei V. Sergeev and David Z. Goodson

The divergent Rayleigh-Schrodinger perturbation expansions for energy eigenvalues of cubic, quartic, sextic and octic oscillators are summed using algebraic approximants. These approximants are generalized Pade 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.

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Calculate Rayleigh - Schrodinger perturbation series for the quartic, cubic, sextic, octic anharmonic oscillators and the Barbanis potential (two-dimensional anharmonic oscillator) using Mathematica programs.

Details that were not included in a paper (Mathematica programs and many figures)

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