Convergence of Scaled Delta Expansion: Anharmonic Oscillator

TL;DR

This paper proves the convergence of the linear delta expansion for the anharmonic oscillator's energy levels under specific scaling of the trial frequency with the expansion order, extending previous results.

Contribution

It establishes rigorous convergence conditions for the delta expansion with a particular scaling of the trial frequency, including the critical case previously discussed in literature.

Findings

Convergence occurs for 1/3<γ<1/2 with appropriate scaling.

Convergence also occurs at γ=1/3 if C exceeds a critical value.

The critical case matches earlier discussions by Seznec, Zinn-Justin, Duncan, and Jones.

Abstract

We prove that the linear delta expansion for energy eigenvalues of the quantum mechanical anharmonic oscillator converges to the exact answer if the order dependent trial frequency $\Omega$ is chosen to scale with the order as $\Omega=CN^\gamma$; $1/3<\gamma<1/2$, $C>0$ as $N\rightarrow\infty$. It converges also for $\gamma=1/3$, if $C\geq\alpha_c g^{1/3}$, $\alpha_c\simeq 0.570875$, where $g$ is the coupling constant in front of the operator $q^4/4$. The extreme case with $\gamma=1/3$, $C=\alpha_cg^{1/3}$ corresponds to the choice discussed earlier by Seznec and Zinn-Justin and, more recently, by Duncan and Jones.