A Convergent Renormalized Strong Coupling Perturbation Expansion

The Rayleigh-Schrr odinger perturbation series for the energy eigenvalue of an anharmonic oscillator deened by the Hamiltonian ^ H (m) () = ^ p 2 + ^ x 2 + ^ x 2m with m = 2; 3; 4;. .. diverges quite strongly for every 6 = 0 and has to summed to produce numerically useful results. However, a divergent weak coupling expansion of that kind cannot be summed eeectively if the coupling constant is 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. which is a power series in a new eeective 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 suuciently large. Moreover, there is strong evidence that it converges for all physically relevant 2 0; 1). The coeecients of the new expansion are deened by divergent series which can be summed eeciently with the help of a sequence transformation which uses explicit remainder estimates E.