The perturbation technique within the framework of the asymptotic iteration method is used to obtain large-order shifted 1/N expansions, where N is the number of spatial dimensions. This method is contrary to the usual Rayleigh-Schr\"{ o}dinger perturbation theory, no matrix elements need to be calculated. The method is applied to the Schr\"{ o}dinger equation and the non-polynomial potential $V(r)=r^2+\frac{ b r^2}{ (1+cr^2)}$ in three dimensions is discussed as an illustrative example.
Categories: quant-ph