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

We present an approximative calculation of the ground-state energy forthe anisotropic oscillator with a potentialV(x,y)=\frac{{\omega}^2}{2} (x^2+y^2)+\frac{g}{4} \left[x^4+2(1-\delta)x^2 y^2+y^4 \right].Using an instanton solution for the isotropic limit $\delta = 0$,we obtain the imaginary part of the ground-state energy for smallnegative $g$ asa series expansion in the anisotropy parameter $\delta$. Fromthis, the large-order behavior of the $g$-expansions accompanying eachpower 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 theisotropic limit. This approximation is found to beexcellent in a rather wide region of $\delta$ around $\delta = 0$.Special attention is devoted to the immediate vicinity of the isotropic point.Using a simple model integral we show that the large-order behaviorof an $\delta$-dependent series expansion in $g$ undergoes a crossoverfrom an isotropic to an anisotropic regime as the order $k$ of the expansioncoefficients passes the value $k_{{\rm cross}} \sim 1/ |{\delta}|$.

AWI Organizations > Biosciences > BioGeoScience

**Eprint ID**