On the quartic anharmonic oscillator and the Padé-approximant averaging method

Abstract

For the quantum quartic anharmonic oscillator with the Hamiltonian [Formula: see text], which is one of the traditional quantum-mechanical and quantum-field-theory models, the summation of its factorially divergent perturbation series is studied on the basis of the proposed method of the averaging of the corresponding Padé approximants. Thus, applying proper averaging weight function, we are able for the first time to construct the Padé-type approximations that possess correct asymptotic behavior at infinity with a rise of the coupling constant [Formula: see text]. The approach gives very essential theoretical and applicatory-computational advantages in applications of the given method. The convergence of the utilized approximations is studied and the values for the ground state energy [Formula: see text] of the anharmonic oscillator are calculated by the proposed method for a wide range of variation of the coupling constant [Formula: see text]. In addition, we perform comparative analysis of the proposed method with the modern Weniger delta-transformation method and show insufficiency of the latter to sum the divergent perturbation series in the region of the superstrong coupling [Formula: see text].