Convergence of scaled delta expansion: Anharmonic oscillator

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 Ω is chosen to scale with the order as Ω = CN^γ; 1/3 < γ < 1/2, C > 0 as N → ∞. It converges also for γ = 1/3, if C ≥ α_cg^1/3, α_c ≃ 0.570875, where g is the coupling constant in front of the operator q⁴/4. The extreme case with γ = 1/3, C = γ_cg^1/3 corresponds to the choice discussed earlier by Seznec and Zinn-Justin and, more recently, by Duncan and Jones.