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 $N\rightarrow\infty$. It converges also for $γ=1/3$, if $C\geqα_c g^{1/3}$, $α_c\simeq 0.570875$, where $g$ is the coupling constant in front of the operator $q^4/4$. The extreme case with $γ=1/3$, $C=α_cg^{1/3}$ corresponds to the choice discussed earlier by Seznec and Zinn-Justin and, more recently, by Duncan and Jones.

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