Large-order shifted 1/N expansions through the asymptotic iteration method

TL;DR

This paper introduces a novel perturbation technique using the asymptotic iteration method to derive large-order shifted 1/N expansions for quantum systems, avoiding matrix element calculations and applied to specific potentials.

Contribution

It develops a new perturbation approach based on the asymptotic iteration method for large-order 1/N expansions, differing from traditional methods by eliminating the need for matrix element calculations.

Findings

Successfully applied to Schrödinger equation with non-polynomial potential

Provides a practical alternative to Rayleigh-Schrödinger perturbation theory

Demonstrates effectiveness for large-order expansions in quantum mechanics

Abstract

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.