Effective characteristic polynomials and two-point Padé approximants as summation techniques for the strongly divergent perturbation expansions of the ground state energies of anharmonic oscillators

Abstract

Pad\'e approximants are able to sum effectively the Rayleigh-Schr\"odinger perturbation series for the ground state energy of the quartic anharmonic oscillator, as well as the corresponding renormalized perturbation expansion [E. J. Weniger, J. \ifmmode \check{C}\else \v{C}\fi{}\'{\i}\ifmmode \check{z}\else \v{z}\fi{}ek, and F. Vinette, J. Math. Phys. 34, 571 (1993)]. In the sextic case, Pad\'e approximants are still able to sum these perturbation series, but convergence is so slow that they are computationally useless. In the octic case, Pad\'e approximants are not powerful enough and fail. On the other hand, the inclusion of only a few additional data from the strong coupling domain [E. J. Weniger, Ann. Phys. (NY) (to be published)] greatly enhances the power of summation methods. The summation techniques, which we consider, are two-point Pad\'e approximants and effective characteristic polynomials. It is shown that these summation methods give good results for the quartic and sextic anharmonic oscillators, and, even in the case of the octic anharmonic oscillator, which represents an extremely challenging summation problem, two-point Pad\'e approximants give relatively good results. \textcopyright{} 1996 The American Physical Society.