A Convergent Renormalized Strong Coupling Perturbation Expansion for the Ground State Energy of the Quartic, Sextic, and Octic Anharmonic Oscillator

Abstract

The Rayleigh–Schrödinger perturbation series for the energy eigenvalue of an anharmonic oscillator defined by the HamiltonianĤ(m)(β)=p2+x2+βx2mwithm=2, 3, 4, … diverges quite strongly for everyβ≠0 and has to summed to produce numerically useful results. However, a divergent weak coupling expansion of that kind cannot be summed effectively if the coupling constantβis large. A renormalized strong coupling expansion for the ground state energy of the quartic, sextic, and octic anharmonic oscillator is constructed on the basis of a renormalization scheme introduced by F. Vinette and J. Čı́žek [J. Math. Phys.32(1991), 3392]. This expansion, which is a power series in a new effective coupling constant with a bounded domain, permits a convenient computation of the ground state energy in the troublesome strong coupling regime. It can be proven rigorously that the new expansion converges if the coupling constant is sufficiently large. Moreover, there is strong evidence that it converges for all physically relevantβ∈[0,∞). The coefficients of the new expansion are defined by divergent series which can be summed efficiently with the help of a sequence transformation which uses explicit remainder estimates[E. J. Weniger,Comput. Phys. Rep.10(1989), 189].