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

For the quantum quartic anharmonic oscillator with the Hamiltonian H = 1/2 (p(2) + x(2)) + lambda x(4), 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 Pade approximants. Thus, applying proper averaging weight function, we are able for the first time to construct the Pade-type approximations that possess correct asymptotic behavior at infinity with a rise of the coupling constant lambda. 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 E-0(lambda) of the anharmonic oscillator are calculated by the proposed method for a wide range of variation of the coupling constant lambda. 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 lambda greater than or similar to 5.

Automated summary

This paper investigates the problem of infinite series divergence in the quantum quartic anharmonic oscillator and proposes an average Pade approximant method to solve it. The method is able to obtain approximations with correct asymptotic behavior as the coupling constant increases. The study demonstrates that the method has significant theoretical and computational advantages in applications, and calculates and analyzes the ground state energy of the anharmonic oscillator extensively.