Resummation of anisotropic quartic oscillator. Crossover from anisotropic to isotropic large-order behavior

We present an approximative calculation of the ground-state energy for the anisotropic anharmonic oscillator Using an instanton solution of the isotropic action $\delta = 0$, we obtain the imaginary part of the ground-state energy for small negative $g$ as a series expansion in the anisotropy parameter $\delta$. From this, the large-order behavior of the $g$-expansions accompanying each power of $\delta$ are obtained by means of a dispersion relation in $g$. These $g$-expansions are summed by a Borel transformation, yielding an approximation to the ground-state energy for the region near the isotropic limit. This approximation is found to be excellent in a rather wide region of $\delta$ aro…