Convergence of Scaled Delta Expansion: Anharmonic Oscillator

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 $\Omega =C{N}^{\gamma }$; $1/3<\gamma <1/2$, $C>0$ as $N\to \infty$. It converges also for $\gamma =1/3$, if $C\ge {\alpha }_c{g}^{1/3}$, ${\alpha }_c\simeq 0.570875$, where $g$ is the coupling constant in front of the operator $q^4/4$. The extreme case with $\gamma =1/3$, $C={\alpha }_c{g}^{1/3}$ corresponds to the choice discussed earlier by Seznec and Zinn-Justin and, more recently, by Duncan and Jones.

Comments: 37 pages (with 11 figures uuencoded at the end of the file,to be stripped off), GEF-Th-7/1994