School of Mathematical and Statistical Sciences Faculty Publications and Presentations
Effective characteristic polynomials and two-point Pade approximants as summation techniques for the strongly divergent perturbation expansions of the ground state energies of anharmonic oscillators
Pade approximants are able to sum effectively the Rayleigh-Schrodinger perturbation series for the ground state energy of the quartic anharmonic oscillator, as well as the corresponding renormalized perturbation expansion [E.J. Weniger, J. Cizek, and F. Vinette, J. Math. Phys. 34, 571 (1993)]. In the sextic case, Pade approximants are still able to sum these perturbation series, but convergence is so slow that they are computationally useless. In the octic case, Pade 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. (N.Y.) (to be published)] greatly enhances the power of summation methods. The summation techniques that we consider are two-point Pade 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 Pade approximants give relatively good results.
Čížek, Jiří, Ernst Joachim Weniger, Paul Bracken, and Vladimír Špirko. 1996. “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.” Physical Review E 53 (3): 2925–39. https://doi.org/10.1103/PhysRevE.53.2925.
Physical Review E
© 1996 The American Physical Society. Original published version available at https://doi.org/10.1103/PhysRevE.53.2925