A CONVERGENT RENORMALIZED STRONG-COUPLING PERTURBATION EXPANSION FOR THE GROUND-STATE ENERGY OF THE QUARTIC, SEXTIC, AND OCTIC ANHARMONIC-OSCILLATOR

The Rayleigh-Schrodinger perturbation series for the energy eigenvalue of an anharmonic oscillator defined by the Hamiltonian (H) over cap(( m))(beta) = (p) over cap(2) + (x) over cap(2) + beta (x) over cap(2m) with m = 2, 3, 4, ... diverges quite strongly for every beta not equal 0 and has to summed to produce numerically useful results. However, a divergent weak coupling expansion of that kind cannot be summed effec tively if the coupling constant beta is large. A renormalized strong c oupling expansion for the ground state energy of the quartic, sextic, and octic anharmonic oscillator is constructed on the basis of a renor malization scheme introduced by F. Vinette and J. Cizek [J. Math. Phys . 32 (1991), 3392]. This expansion, which is a power series in a new e ffective coupling constant with a bounded domain, permits a convenient computation of the ground state energy in the troublesome strong coup ling regime. It can be proven rigorously that the new expansion conver ges if the coupling constant is sufficiently large. Moreover, there is strong evidence that it converges for all physically relevant beta is an element of [0, infinity). The coefficients of the new expansion ar e defined by divergent series which can be summed efficiently with the help of a sequence transformation which uses explicit remainder estim ates [E. J. Weniger, Comput. Phys. Rep. 10 (1989), 189]. (C) 1996 Acad emic Press, Inc.