Continued fractions and Rayleigh–Schrödinger perturbation theory at large order

Abstract

Concern with the continued fraction representations of divergent Rayleigh–Schrödinger perturbation expansions in quantum mechanics is expressed. The following relation between the large-order behavior of the continued fraction coefficients cn and the perturbation series coefficients E(n) is shown to exist: If E(n) ∼(−1)n+1Γ( pn+a), p=0,1,2,..., as n→∞, then cn=O(np) as n→∞. The case p=1 is studied in detail here, using the problems of the quartic anharmonic oscillator and the hydrogen atom in a linear radial potential as illustrative examples. For p=1 the asymptotics of the cn are shown to be linked to the infinite field limit E(λ)∼F(0)λα, predicting α and providing convergent estimates of F(0).